r    ^^ 


*\   ^^p^   /      \      A 


\.  J" 


AN 


ELEMENTARY  ALGEBRA : 


DESIGI^ED    AS 

I 

AN  INTRODUCTION   TO   A  THOROUGH   KNOWLEDGE 

OF  ALGEBRAIC  LANGUAGE,  AND  TO   GIVE 

BEGINNERS  FACILITY  IN  THE  USE  OF 

ALGEBRAIC  SYMBOLS. 


CHARLES   S.  VENABLE,  LLD., 

PfiOPESSOR  OF  Mathematics  in  the  University  o?  Yirginia  ;  Author  op  "  First 

Lessons   in   Numbers,"    "Mental   Arithmetic,"   "Practical 

Arithmetic,"   and    "Higher  Arithmetic." 


UNIVERSITY    PUBLISHING    COMPANY, 

NEW    YORK    AND    BALTIMORE. 

1872. 


or  THE 


VNIVERRITY 


urn 
V4. 


/^ 


Entered  according  to  Act  of  Congress,  in  the  year  186f), 

By  the  university  .^L'BLISHING  COMPANY, 

m  the  Clerk's  ofScQ  rf  the  District  Co'irt  of  the  United  States  for  the  Southern 

i;:slrict  of  New  \  ork. 


PREFACE. 


The  present  Elementary  Algebra  has  been  prepared  with 
a  view  to  enable  the  beginner  to  obtain  a  thorough  knowl- 
edge of  Algebraic  Language,  and  to  acquire  an  early  facility 
in  the  use  of  Algebraic  Symbols.  The  translation  of  Ex  g- 
lish  into  the  symbolical  language  of  Algebra,  and  the  inter- 
pretation of  Algebraic  Symbols  by  arithmetical  operations 
are  made  prominent  from  the  beginning.  Throughout  the 
work  I  have  endeavored,  in  the  Algebraic  operations  and 
solutions  of  problems,  to  present  examples  of  elegance  and 
conciseness  in  the  transformation  of  Algebraic  expressions. 
I  am  convinced  by  long  observation  that  the  difficulties  of 
students  in  their  more  advanced  mathematical  studies  are 
greatly  enhanced  by  their  want  of  knowledge  of  Algebra  as 
a  Language,  and  their  want  of  facility  m  the  transformation 
and  combination  of  expressions  in  the  solution  of  problems. 
These  tM7igs  form  tlie  dasis  of  any  thorough  Icnowledge  of 
Algebraic  Analysis,  and  should  he  learned  ivellin  the  begin- 
ning. The  demonstrations  are,  I  think,  clear  and  easily 
rntelligible  to  the  young  student.  The  examples  for  exercise 
are  numerous. 

In  addition  to  the  fundamental  Algebraic  operations  on 
Entire  Quantities  and  Fractions,  Evolution,  Surds,  Equa- 


4  PEEFACE. 

tions,  Arithmetical  and  Geometrical  Progressions,  and  Pro- 
portion,  I  have  treated  in  an  elementary  manner  the  subjects 
of  Fractional  Exponents,  Permutations  and  Combinations, 
the  Binomial  Theorem  for  whole-number  exponents,  Har- 
monical  Progression,  Theory  of  dotation,  and  Logarithms. 
I  am  convinced  by  long  experience  that  it  is  important  to 
present  these  subjects  to  the  young  student  in  a  simple  and 
practical  manner  before  he  comes  in  contact  with  them  in 
their  greater  extensions  and  more  difficult  applications.  In 
the  preparation  of  this  book  I  have  consulted  many  of  those 
works  which  give  a  view  of  the  progress  and  improvement  in 
elementary  instruction  in  Algebra.  But  three  English  works 
— Todhunter's  Algebra  for  Beginners,  Colenso's  Algebra, 
and  Lund^s  Wood's  Algebra — are  made  the  basis  of  the 
worK.  The  demonstrations  of  Wood,  (a  standard  of  more 
than  half  a  century,)  are  singularly  clear  and  simple,  while 
those  of  Colenso  are  models  of  elegance  and  brevity.  Tod- 
hunter's  illustrations  are  clear  and  copious.  The  examples 
have  been  selected  mainly  from  the  above  authors,  many  of 
them  having  been  taken  by  them  from  the  Cambridge  Ex- 
amination papers.  I  have  also  used  Lund's  Easy  Algebra, 
Bobillier's  "Principes  d'Algebre,"  Kitt's  Problemes  d'Al- 
gebre,  and  Wrigley's  Collection  of  Problems. 

University  of  Virginia, 
Aug.  I,  1869. 


CONTENTS. 


Pagb 

I.    Principal  Signs 7 

II.  Factor — Coefficient — Power — Terms         .     ii 

III.  Remaining  Signs— Brackets     .        .        .         i6 

IV.  Change  of  Order  of  Terms— Like  Terms      20 
V.     Addition 24 

VI.     Subtraction 28 

VII.     Brackets 33 

VIII.     Multiplication 38 

IX.  General  Results  of  Multiplication     .         45 

X.     Division 52 

yXI.     Factors 61 

XII.  Greatest  Common  Divisor    .        •       •        .    (^^ 

XIII.  Least  Common  Multiple  ....         "](> 

XIV.  Fractions 80 

XV.  Reduction  of  Fractions  ....         84 

XVI.  Addition  and  Subtraction  of  Fractions       88 

XVII.  Multiplication  of  Fractions  ...         94 

XVIII.     Division  of  Fractions 98 

XIX.  Complex  Fractions,  and  other  Results       ioi 

XX.     Involution 105 

XXI.  Evolution    .        .    '    .        .        .        .        .        no 

XXII.  Simple  Equations 128 

XXIII.  Simple  Equations — continued         .        .       137 

XXIV.  Problems  Solved  by  Simple  Equations      .  145 
XXV.  Problems — continued        .        .        .        .       153 


6  CO]SrTEis-TS. 

XXVI.     Simultaneous    Equations    of    the    First 

Degree 164 

XXVII.     Problems  Solved  by  Simultaneous  Equa- 
tions OF  THE  First  Degree         .        .        176 

XXVIII.     Indices 184 

XXIX.     Surds    ...'.....        191 

XXX.     Quadratic  Equations 200 

XXXI.    Equations    which    may    be    solved    like 

Quadratic  Equations    .        .        .        .       212 
XXXII.     Problems     which     lead     to     Quadratic 
Equations    containing    One    Unknown 
Quantity      ....        ...  217 

XXXIII.  Simultaneous  Equations  involving  Quad- 

ratics      .        .        .        .        .        .        .221 

XXXIV.  Ratio 230 

XXXV.    Proportion 233 

XXXVI.  Arithmetical  Progression    ....  240 

XXXVII.  Geometrical  Progression        ...       245 

XXXVIII.  Harmonical  Progression       ....  252 

XXXIX.  Permutations  and  Combinations    .        .       254 

XL.  Binomial  Theorem  .        .        .        .        .        .  263 

XLI.    Scales  of  Notation 269 

XLII.     Logarithms      .        . 275 

Answers  to  Examples       .        .        .       •       284 


ELEMENTARY    ALGEBRA. 


I.    The  Peincipal  Signs. 


1.  Algebra  is  the  science  in  which  Ave  reason  about  num- 
bers with  the  aid  of  letters  to  denote  the  numbers,  and  of 
certain  signs  to  denote  the  operations  performed  on  the 
numbers,  and  the  relations  of  the  numbers  to  each  other. 
These  letters  and  signs  are  called  Algehraic  Symbols, 

2.  Quantity  signifies  anything  which  admits  of  increase, 
or  diminution.  The  word  quantity  is  often  used  with  the 
same  meaning  as  numler, 

3.  The  sign  +  placed  before  a  number  denotes  that  this 
number  is  to  be  added.  Thus  a  ^1)  denotes  that  the  num- 
ber represented  by  h  is  to  be  added  to  the  number  repre- 
sented by  a.  If  a  represent  9  and  t)  represent  3,  then  a  +  5 
represents  12.  The  sign  -f  is  called  the  ])lu8  sigti,  and 
a  +  Z>  is  read  "  a  plus  b." 

4.  The  sign  —  placed  before  a  number,  denotes  that  the 
number  is  to  be  subtracted.  Thus  a  —  b  denotes  that  the 
number  represented  by  Z>,  is  to  be  subtracted  from  the  num- 
ber represented  by  a.  If  a  represent  9  and  b  represent  3, 
then  a  —  b  represents  6.  The  sign  —  is  called  the  minus 
sign,  and  a  ~  ^  is  read  thus,  "  a  minus  b." 

5.  Similarly,  a  +  b  ■\-  c  denotes  that  we  are  to  add  b  to  a, 
and  then  add  c  to  the  result ;    a-\-b  —  c  denotes   that  we 

Define  Algebra  ;    Algebraic  Symbols  ;    Quantity,     How  is  Addition  indicated  ? 
Subtraction  ? 


8  ELEMENTARY    ALGEBRA 

are  to  add  h  to  a,  and  then  subtract  c  from  the  result  ]  a  —  l) 
+  c  denotes  that  we  are  to  subtract  1)  from  a,  and  then  add  c 
to  the  result ;  a  —  h  —  c  denotes  that  we  are  to  subtract  h 
from  a,  and  then  subtract  c  from  the  result. 

6.  The  sign  =  denotes  that  the  numbers  between  which  it 
is  placed  are  equal.  Thus  a  =  l  denotes  that  the  number 
represented  by  a  is  equal  to  the  number  represented  by  l. 
And  a  -{-h  —  c  denotes  that  the  sum  of  the  numbers  repre- 
sented by  a  and  h  is  equal  to  the  number  represented  by  c; 
so  that  if  a  represent  9  and  ^  represent  3,  then  c  must 
represent  12.  The  sign  —  is  called  the  sign  of  equality, 
and  a  =  Z>  is  read  thus,  "  a  equals  b,"  or  a  is  equal  to  h, 

7.  The  sign  X  denotes  that  the  numbers  between  which  it 
stands  are  to  be  multiplied  together.  Thus  a  Xl)  denotes 
that  the  number  represented  by  a  is  to  be  multiplied  by  the 
number  represented  by  b.  If  a  represent  9  and  b  represent 
d,  then  a  X  b  represents  27.  The  sign  X  is  called  the  sign 
of  multiplication,  and  ^  X  ^  is  read  thus,  "  a  into  b/^  or  "  a 
multiplied  ly  b.^^  Similarly,  aXbX  c  denotes  the  product 
of  the  numbers  represented  by  a,  h,  and  c, 

8.  Sometimes  a  point  is  used  instead  of  the  sign  X .  Thus 
a.  I)  instead  of  a  X  ^.  Both  of  these  signs  are,  however,  often 
omitted  for  the  sake  of  brevity ;  thus  a  h  is  used  instead  of 
aXh,  and  has  the  same  meaning.  So  also,  ahc  i^  used  in- 
stead of  aXb  X  c,  or  a.b.c,  and  has  the  same  meaning, 
Nor  is  either  the  point  or  the  sign  X  necessary  between  a 
numbei  expressed  by  a  figure  and  a  number  expressed  by  a 
letter ;  so  that  3  a  is  used  instead  of  3X  a,  and  has  the  same 
meaning.  The  sign  of  multiplication  must  not  be  omitted 
between  numbers  expressed  in  the  ordinary  way  by  figures. 
Thus  45  cannot  be  used  to  represent  the  product  of  4  and  5, 
because  the  meaning,  forty-five,  has  already  been  assigned  to 
45.    Nor  can  the  point  be  used  between  figures  to  express  a 

How  is  Equality  indicated  ?    Multiplication  ? 


PKIIS^CIPAL    SIGKS.  9 

product,  witliout  prodiiciDg  confusion,  as,  for  example,  to 
4 . 5  has  already,  in  Arithmetic,  been  assigned  the  meaning 
4  +  Y^Q- ;  still,  4 . 5  is  sometimes  written  for  4x5. 

0.  The  sign  -i-  denotes  that  the  quantity  which  stands 
defore  it  is  to  be  divided  by  the  quantity  which  follows  it. 
Thus  a-^h  denotes  that  the  number  represented  by  a  is  to 
be  divided  by  the  number  represented  by  I.  If  a  represent 
12,  and  1)  represent  4,  then  a  -^l  represents  3.  The  sign  -^ 
is  called  the  sign  of  division,  and  « -f-  Z>  is  read  thus,  "  a 
divided  hy  b,"  or  briefly,  "  a  ly  b." 

But  most  frequently  to  express  division  the  dividend  is 

placed  over  the  divisor,  with  a  line  betw^een  them.      Thus  -^ 

is  used  for  a-^l,  and  has  the  same  meaning. 

10.  A  number  or  quantity  expressed  by  Algebraic  Sym- 
bols is  called  an  Algebraic  exinession,  or  briefly,  an  expres- 
sion, 

11.  We  shall  now  give  some  examples  as  an  exercise  in  the 
use  of  the  symbols  which  have  been  explained ;  these  exam- 
ples consist  in  finding  the  numerical  values  of  certain  Alge- 
braic expressions,  and  in  finding  the  Algebraic  expression 
for  certain  quantities  expressed  in  ordinary  language. 

Suppose  a  =  1,  l)  —  2,  c  —  'd,d  —  6,  e  =  6,  f=  0.     Then 

7«  +  3Z'-2  6?+/=74-6-10  +  0  =  13-10  =  3. 

2  ad +  Shc-ae  +  df  =4:  + 4.8 -6  +  0  =  62-6  =  4^6. 

4:ac  ,  lObe     de     12  ,120     30      ^  ,   _      ,^ 
b  cd       ac      2       lo       3 

14  -  10  =  4. 

4:c  +  6e_  12  +  30  _  42__ 
d-b    ""    5-2    ~  3  ~      ' 

How  is  Division  indicated?    What  is  an  Algebraic  Expression ? 


10  eleme:n^tary   algebra. 

Examples — 1. 

li  a=l,  b^  Vf  c=3,  cl=4:,  e=5,f  =  0y  find  the  numerical 
values  of  the    jllowing  expressions  : 

1.  9a+2I)+3c-2f.  2.     4:e-3a-dI}-\-5c. 

3.  7ae-{-3I)c+dcl-af.  4.     8adc-hcd+9cde-def. 

5.  ahcd-habce-habde  +  acde+dcde,    6.  -7- h h-r . 

0      c      d      e 

4:ac     8bc     6cd  _      12a     6b     20c 

7.    —J j J .  O.        -y 1 ---H —• 

ode  be     cd      de 

^   cde  ,  hbcd    6ade  ^.      ^.77-  ^^de 

9.  — 7-H = — .  10.     7e-hbcd — ^ — . 

ab       ae        be  2ao 

2a+6b    Sb-h2c    a+b+c  +  d  b+c-{-3e 

c  d  2e  e+c—d 

-Q   a  +  c    b-\-d  .  c+e  ^.      a+b+c  +  d-\-e 

c  —  a    d—b    e—c  e—d  +  c—b-\-a 

15.  What  is  the  difference   between  ~-\ —  and  -j-'  when 

be  be 

a  =  4,  6  =  5,  c  =  10  ?  Ans.  5^. 

16.  A  person  who  possessed  a  fortune,  expressed  by  x,  re- 
ceives by  inheritance  two  sums,  a  and  b.  Express  his  whole 
property. 

17.  An  estate  is  divided  among  three  heirs:  the  second  ob- 
tains a  dollars  more  than  the  first,  and  the  third  b  dollars 
more  than  the  second.  Express  the  value  of  the  estate,  tlie 
share  of  the  first  heir  being  x. 

18.  A  man  who  possessed  x  dollars,*  lost  a  of  them.  How 
many  has  he  left  ? 


FACTOR. — COEFFICIEKT. — POWEK. — TERMS.  1 1 

19.  The  Slim  of  two  quantities  being  represented  by  s,  if 
one  of  these  quantities  is  expressed  by  x,  what  will  be  the 
other  ? 

20.  A  debtor  set  out  to  pay  his  creditor  a  part,  a^  of  his 
debt  X ;  but  on  the  way  he  met  another  person,  to  whom  he 
gave  a  sum  Z>  and  carried  the  remainder  to  his  creditor.  Ex- 
press what  remains  of  the  debt. 

21.  The  three  figures  of  a  number  are  such  that  the  tens 
figure  exceeds  the  units  figure  by  2,  and  the  hundreds  figure 
exceeds  the  tens  figure  by  3.  What  is  the  sum  of  the  figures, 
X  being  the  units  figure  ? 

22.  Express  45.6  by  means  of  algebraic  signs. 

23.  The  figures  of  a  number  in  the  hundreds,  tens,  and 
units  place  respectively  are  x,  y,  z.  Express  the  number. 
Express  the  number  with  the  figures  reversed. 

24.  A  w^orkman  whose  wages  were  x  dollars  a  day,  receives 
during  n  days  an  increase  of  wages  amounting  to  h  dollars  a 
day.     Express  the  amount  of  his  wages  for  this  time. 

25.  Express  the  number  which  divided  by  11  gives  5  for  a 
remainder,  x  being  the  quotient. 

26.  A  man  travels  uniformly  a  distance  s  in  a  number 
t  of  hours.      AYhat  is  his  rate  of  travel  per  hour  ? 

27.  Two  fountains  fill  a  reservoir,  one  in  a  number  of 
hours  represented  by  x,  the  other  in  3  hours  more.  What 
part  of  the  reservoir  does  each  one  fill  in  one  hour  ? 

28.  A  number  N  divided  by  d  gave  for  remainder  the 
number  r.     Express  the  quotient. 

II.    Factor — Coefficient — Power — Terms. 

12.  When  one  number  consists  of  the  product  of  two  or 
more  numbers,  each  of  the  latter  is  called  a  factor  of  the 
product.     Thus,  for  example,  2  X  3  X  5  =  30 ;  and  each  of 

Define  Factor. 


1^  ELEMENTARY    ALGEBRA. 

the  numbers  2,  3,  and  5  is  a  factor  of  the  product  30.  Or  we 
may  regard  30  as  the  product  of  the  two  factors  2  and  15,  or 
as  the  product  of  the  two  factors  3  and  10.  So  also  we  may 
consider  4  a  Z>  as  the  product  of  the  three  factors  4,  a,  and  ^, 
or  of  the  two  factors  4  a  and  b;  or  as  the  product  of  the  two 
factors  4  and  ad;  or  as  the  product  of  the  two  factors  4 d 
and  a, 

13.  When  a  number  consists  of  the  product  of  two  fac- 
iTors,  each  factor  is  called  the  coefficient  of  the  other  factor ; 
so  that  coefficient  means  co-factor.  Thus,  considering  4: ad 
as  the  product  of  4  and  a  d,  we  call  4  the  coefficient  of  a  h, 
and  ab  the  coefficient  of  4;  and  considering  4a^  as  the 
product  of  4:  a  and  d,  we  call  4  a  the  coefficient  of  b,  and  I? 
the  coefficient  of  4  a.  In  practice,  the  name  coefficient  is  ap- 
plied to  the  factor  which  precedes  the  other,  and  usually  the 
first  factor  is  called  the  coefficient.  When  this  first  factor  is 
a  number  expressed  by  a  figure,  it  is  called  the  oiumerical 
coefficient.  Thus  4  is  the  numerical  coefficie7it  of  ab  in  the 
expression  4:al).  Since  1  is  a  factor  of  every  product,  when 
no  numerical  coefficient  is  written  before  an  algebraic  ex- 
pression 1  is  always  understood  as  its  7iumerical  coefficient. 
Thus  1  is  the  numerical  coefficient  of  the  expressions,  a  h,  a, 
abc,  Wlien  we  use  one  of  the  letters  of  a  product  as  a  coef- 
ficient it  is  called  a  literal  coefficient.  Thus  in  the  expres- 
sion a  X,  a  is  the  literal  coefficient  of  x, 

14.  When  all  the  factors  of  a  product  are  equal,  the 
product  is  called  a  poiver  of  that  factor.  Thus  7  X  7  is 
called  the  second  poiver  of  7;  7  X  7  X  7  is  called  the  third 
foioer  of  7 ;  7  X  7  X  7  X  7  is  called  the  fourtli  power  of  7, 
and  so  on.  In  like  manner,  «  X  <^  is  called  the  second  poiver 
of  « ;  aX  aX  a\s  called  the  third  power  ofa;  aXaXaXa 
is  called  ihQ  fourth  power  of  a,  and  so  on.  And  a  itself  is 
called  the  first  p)0wer  of  a. 

Coefficient,  numerical  and  literal.    Power. 


FACTOE — COEFFICIEXT — POWEE — TEEMS.  1 3 

15.  Instead  of  writing  all  the  equal  factors,  we  express  a 
power  more  briefly  by  writing  the  factor  once,  and  placing 
over  it,  and  a  little  to  the  right,  the  number  which  indicates 
how  often  the  factor  is  to  be  repeated.  Thus  c^  is  used  to  de- 
note ^  X  «  ;  ci^  denotes  a  X  aX  a]  a^  denotes  a  X  a  X  a  X  a; 
3a^ b^ c^ d  denotes  oaaaahl)h  ccd,  ^ndi  so  on.  And  a^  may 
be  used  to  denote  the  first  power  of  a,  that  is,  a  itself;  so 
that  a^  has  the  same  ^meaning  as  a.  The  number  thus  placed 
over  another  to  indicate  how  many  times  the  latter  occurs  as 
a  factor  in.  a  power,  is  called  an  index  of  the  jpoiver,  or  an  ex- 
ponent  of  the  i^oiver^  or  briefly  an  index  or  exponent.  Tlius, 
for  example,  in  a^  the  exponent  is  3 ;  in  a""  the  exponent  is 
n ;  in  2""  the  exponent  is  m, 

16.  The  second  power  of  a,  that  is  a^,  is  usually  called  the 
square  oi  a  oy  a  squared ;  and  a^  is  often  called  the  cuhe  of 
a  or  a  cubed.  For  the  powers  higher  than  these  tliere  are  no 
similar  words  in  use ;  a'^  is  read  thus,  "  a  to  the  fourth 
potver/'  or  "  a  to  the  fourth  f  a"^  is  read  "  a  to  the  7nth." 

17.  The  student  must  distinguish  carefully  between  a 
coefficient  and  an  exponent.  Thus  3  c  means  three  times  c, 
and  m  a  stands  for  m  times  a.  Here  3  and  m  are  coefficients. 
But  c^  means  c  times  c  times  c ;  and  c"^  stands  for  c  times  o 
times  c  times  c  and  so  on  ...  .  times  c. 

18.  If  an  expression  contain  no  parts  connected  by  the 
signs  +  and  — ,  it  is  called  a  simple  expression  or  monomial. 
If  an  expression  contain  parts  connected  by  the  signs  + 
and  — ,  it  is  called  a  compound  expression ;  and  the  parts 
connected  by  the  signs  +  and  —  are  called  terms  of  the  ex- 
pression. Thus  ax,  4: be,  and  6  a""  c"^  are  simple  expressions; 
a^-^-F—c^  a^-{-?ta^b'\-c^,  are  compound  expressions,  of  which 
a^,  F,  and  c\  and  ct\  3  «^  b,  and  c^  are  the  tenns  respectively. 

19.  Let  the  student  distinguish  carefully  between  tenns 

Index  or  Exponent,    a^ .  ^3  •  ^.,1     Difference  between  Coefficient  and  Exponent. 
Monomial  Terms.    Terms  and  Factors. 


14  ELEMEKTARY    ALGEBRA. 

and  factoids,  recollecting  that  factors  are  those  parts  of  an 
expression  which  are  connected  by  Multiplication,  and  terms 
are  those  parts  of  a  compound  expression  which  are  connected 
by  Addition  and  Subtraction. 

Thus  5,  a^y  d,  and  c  are  factors  of  the  expression  6a'^bc; 
while  6cr,  h,  and  c  are  terms  of  the  expression  ba^  -^-1)—  c» 

20.  When  an  expression  consists  of  tivo  terms  it  is  called 
a  Uiiomial  expression,  or  briefly  a  binomial ;  when  it  consists 
of  three  terms  it  is  called  a  triyiomial ;  any  expression  con- 
sisting of  several  terms  is  called  Si  multinomial  or  jwli/nomial 

Thus  2«^ -\-dabc  is  a  binomial  expression ;  a  —  2b  -h  6c 
is  a  trinomial  expression  ;  and  a"^—  ¥+  c^—  4tab  —  e  is  a 
polynomial. 

21.  Each  of  the  letters  or  literal  factors  of  a  term  is  called 
a  dimension  of  the  term,  and  the  number  of  these  literal  fac- 
tors is  called  the  degree  of  the  term.  Thus  2a^b^c  on:  2  a  abb  be 
has  six  dimensions  and  is  said  to  be  of  the  sixth  degree. 
The  numerical  coefficient  is  not  counted;  thus  9a^^*  and 
a^  V  are  of  the  same  dimensions,  namely,  seven  dimensions. 
The  degree  of  a  term,  that  is,  the  number  of  its  dimensions, 
is  evidently  the  sum  of  the  exponents  of  its  literal  factors, 
provided  we  remember  that  when  no  exponent  is  expressed 
the  exponent  1  must  be  understood. 

22.  An  expression  is  said  to  be  Jiomogeneous  when  all  its 
terms  are  of  the  same  degree. 

Thus  '7  a^  -hSa^  -i-  4:abci8  homogeneous,  for  each  term  is 
of  three  dimensions,  that  is,  of  the  third  degree. 

We  shall  now  give  some  more  examples  of  finding  the  nu- 
merical values  of  algebraic  expressions. 

Suppose  a  =  1,  b  =  2,  c  =  d,  d  =  4:,  e  =  5,  /=  0.     Then 

b'  =4,     b'  =  8,     h*  =  16,     b'  =  32. 

Binomial.  Trinomial,  Polynomial.  Dimension  and  Degree  of  Term.  When  is 
an  expression  homogeneous? 


VN{Y£RSrTY   I 


15. 


a'^^^z^lxS^S,  3^V=3X4X9  =  108. 
^  +  c'-7aZ»-f-/'=:  64  +  9-14  +  0  =  59. 


3c^-4^-10 


27-12-10         5      . 
— =  0. 


c'-2c'  +  5c-23     27-18  +  15-23     1 


6^  +  rr     g^-a^_125+64     27-l_189     26_  _ 

e+^        c-a~     5  +  4         3-1  ~   9        2~  ■^^~^' 


Examples — 2. 

If  a  =  1,  Z>  =  2,  c  =  3,  rf  =  4,  e  =  5,  /  =  0,  find  the  numer- 
ical A'alues  of  the  following  expressions. 


1.     «'+Z/'  +  c'+^+e'+/'. 
5.     a'-\-?>a^l^Zah^^l\ 


^V     ^_32 


9      ^'  +  ^'     ^'  +  ^'     e'-cl" 


6. 
8. 


2.    e'-J^  +  c'-JHa'. 

4.     c^-"2c^+4c-13. 
e*-4e'^+6e*J'-4eJH^*. 
2g  +  2     3g-9     e--l 


e-d 


e—2       e+3 


8a^+3^^     4:c''+6b\    c'  +  ^ 

A  v.  o     ,      7  9        "1 


11. 


a'  +  b' 

28 


1  + 


12 


+  - 


a^+^^  +  6''^rf-^^-^*^^V  +  6^-c'-6?'* 


12. 


a'-V^fn-V^crb^-V^ab^-\-b'' 
a^-^?>d'b-\-Zab^V      ' 


13. 


d' 


14. 


^  +  Z>'' 


16  ELEMENTARY    ALGEBRA. 

III.    Kemaining  Signs — Brackets. 

23.  Tlie  sign  >  stands  for  is  greater  tlian,  and  the  sign  < 
denotes  is  less  than,  Tims  a>  h  denotes  that  the  quantity  a 
is  greater  than  the  quantity  h,  and  h  <a  denotes  that  the 
quantity  h  is  less  than  the  quantity  a.  In  this  sign  the 
opening  is  always  turned  toward  the  greater  quantity. 

24.  The  sign  .*.  denotes  then  or  therefore;  the  sign  •.*  de- 
notes since  or  because, 

25.  The  square  root  of  a  quantity  is  that  quantity  whose 
square  or  second  potver  is  equal  to  the  giyen  quantity. 

'  The  cube,  fourth,  &c.  root  of  a  given  quantity  is  that 
quantity  whose  cube,  fourth,  &c.  power  is  equal  to  the  given 
quantity.  Thus  since  49  ==  T,  the  square  root  of  49  is  7; 
also  the  square  root  of  a^  is  a.  In  like  manner,  since 
125  =:  5^  the  cube  root  of  125  is  5 ;  and  so  if  a  =  c^y  the 
cube  root  of  a  is  c, 

26.  The  symbol  used  to  denote  a  root  is  \/  (a  corruption 
of  r,  the  first  letter  of  the  word  radix),  which  with  the  proper 
number  as  index  on  the  left  side  of  it,  a  little  above,  is  set 
before  the  quantity  whose  root  is  expressed. 

Thus  V«'  ^  a,  V64  =  4,  V3125  =  5,  VI  =  1,  VI  =  1,  &c. 
The  index,  however,  is  generally  omitted  in  denoting  the 
square  root ;  thus  v^a  is  written  instead  of  V«. 

Examples — 3. 

1.  v/4"+2v/25  +  3v/49-v/ 64  =  25. 

2.  3n/i6-4v/36  +  2v/9'-\/8T=-15. 

3.  V8"+2Vi25-4Vr4-V64  =  12. 

4.  vr+3Vi6-2V32+3vr=6. 

If  a=2D,  b=9,  c=4:,  d=l,  then 
5.    y/a-{-2^/l;  +  3^/'^  +  4:^/d=2l. 

Explain  the  signs  >  <  .*.  •.•  Square  root  of  a  quantity  ?  Cube  root?  Th« 
Bymbol  used  to  denote  a  root. 


KEilAlXIXG    SIGXS. — BRACKETS.  17 

7.  3Va+2VAb-4:Vrc+Vm=7. 

8.  V5^+2V3b-V2c-h4:Vd=13. 

10.    Vbc+dVacd  —  4:Wd+V^^  —  4:, 

27.  |/^  means  that  the  square  root  of  the  fraction  %  is  to 

be  taken ;  but  — —  means  that  the  square  root  of  a  is  to  be 
divided  by  t. 

Examples. 

1.  What  is  the  difference  between  2\/'^  and  2+  \/^^ 
when  X  is  100  ?  .  Ans.     8. 

2.  What  is  the  difference  between  3  \/"^,  and  V^,  when  x 
is  64  ?  Ans.  20. 

3.  What  is  the  difference  between  \/  a-^h  and  v/"^  +  Z>, 
when  a  stands  for  1,  and  1)  for  8  ?  Ans.     6. 

4.  What  is  the   difference  between  \/%-  and  -— ^,  when 

a  stands  for  16  and  Z>  for  4?  Ans.     1. 

5.  What  is  the  difference  between  v/ «  +  \/y  and  >/^+^^ 
when  a  =  16  and  Z>  =  9?  Ans.     2. 

28.  Brackets,  (),{},[],  are  employed  to  show  that  all  the 
quantities  within  them  are  to  be  treated  as  though  forming 
but  one  quantity.  It  is  of  great  importance  to  notice  care- 
fully the  effect  of  using  them. 

Thus  a—{b—c)  is  not  the  same  as  a—b—c;  for,  in  this 
last,  both  b  and  c  are  subtracted,  whereas  in  the  former  it  is 
only  the  quantity  b  —  c  which  is  subtracted. 

The  use  of  Brackets. 


18  ^  ELEMEKTAEY    ALGEBRA. 

Hence,  if  a  =  4:,  b  =  3,  c  =  1,  we  liaye 

a  — d  — 0  =  4.-3  —  1  =  0,  a—{b  —  c)=4  — 2  =  2; 

2a-'3h  +  2c=S-9+2  =  l,2a-{3b  +  2c)  =  S-ll  =  -3, 

2a-^b-c=S+3-l  =  10,2{a+b)-c=U-l  =  13, 

2{a  +  I)-c)=12. 

If  we  wisli  to  denote  that  the  sum  of  a  and  d  is  to  be  mul- 
tiplied by  Cy  we  write  {a+b)Xc  or  {a  +  I)}Xc,  or  simply 
{a-}-I))c  or  {a  +  d}c;  here  we  mean  that  the  whole  of  a-\-b  is 
to  be  multiplied  by  c.  Now  if  the  brackets  were  omitted  we 
would  have  a  +  be,  which  denotes  that  b  only  is  to  be  multi- 
plied by  c  and  the  result  added  to  a.  Similarly  {a  +  b—c)d 
denotes  that  the  result  expressed  by  a  +  Z*  —  c  is  to  be  multi- 
plied by  d. 

So  also  {a  —  b  +  c)x{d  +  e)  denotes  that  the  result  ex- 
pressed by  a  —  J  +  c  is  to  be  multiplied  by  the  result  ex- 
pressed by  d  +  e.  This  may  also  be  denoted  briefly  thus, 
(a  —  b  +  c)  {d-\-  e);  just  as  a  X  ^  is  shortened  into  a  b. 

So  also  V{a  +  b-\-c)  stands  for  the  square  root  of  the  result 
expressed  hj  a  +  b-\-c. 

So  also  \/{a+b-\-c)  denotes  that  we  are  to  obtain  the  re- 
sult expressed  hj  a+b+c,  and  then  take  the  square  root  of 
this  result. 

So  also  {aby  denotes  abX  ab;  and  (a by  denotes 
abXabXab. 

So  also  {a+b—c)-^{d-\-e)  denotes  that  the  result  expressed 
by  a  -\-  b  —  c  is  to  be  divided  by  the  result  expressed  by 
d-\-e. 

29.  Sometimes  instead  of  using  brackets  a  line  is  drawn 
oyer  the  numbers  which  are  to  be  treated  as  forming  one 
number.      Thus  a—b-\'CXd+e  is  used  with  the  same  mean- 

Vinculum. 


kemai:n'ikg  sigxs — brackets.  19 

ing  as  {a—bi-c)x{d+e),  A  line  used  for  this  purpose  is 
called  a  vincuhun.  So  also  {a+b—c)-^{d-{-e)  may  be  de- 
noted thus,  — 9— — ;  and  here  the  line  between   a  -\-  b  —  c 

and  d-{-  eh  really  a  vinculum  used  in  a  particular  sense. 

Thus,  too,  a  vinculum  from  the  top  of  a  radical  sign  is  fre- 
quently used,  and  \/  a  +  h  +  c  has  the  same  meaning  as 
V{a-\-b-\'c). 

30.  We  have  now  explained  most  of  the  signs  used  in 
Algebra.  It  is  well  to  observe  that  the  word  sign  is  applied 
specially  to  the  two  signs  +  and  — .  Thus  the  expressions 
"changing  the  signs^^  "like  signs,"  and  "unlike  signs"  refer 
exclusively  to  +  and  — . 

31.  We  shall  now  give  some  more  examples  of  finding  the 
numerical  values  of  expressions. 

Suppose  ^  =  1,  5  =  2,  c  =  3,  ^  =  5,  e  =  8.     Then 

V(4c-2^>)=V(12-4)=V(8):ir:2. 

eN/(25+46')--(26Z-^)V(4c-2^)=:::8x4~8X  2=32-16  =  16. 
v/{(6-^)(26-5Z^)}-v/{(8-2)(16-10)}  =  v/(6x6)3=6. 
{(e-^)(^4-c)-(c?-c)(c+a)}(fl^  +  ^)^{3x5-2x4}6 

=  7X6=42. 
V(c''  +  3c'^5  +  3c5'^  +  &')-^v/(^^+Z>^-2a^) 

=  V(27  +  54  +  36  +  8)--v/(l  +  4-4)=V(125)-^l=5. 

Examples — 4. 

If  a  =  1,  5  =  2,  c  =  3,  J  =  5,  e  =  8,  find  the  numerical 
values  of  the  following  expressions. 

The  word  Sign^  how  applied? 


20  ELEMEXTARY    ALGEBRA. 

1.     a{h  +  c).  2.     b{c  +  d):  3.     c{e~d). 


4.     b^a'+e'-c').      5.     c'ie'-F-c'),       6. 


cr+b' 


7.  ^"^^-.        8.     n/(3&c^).         9.     V{2b+U+6e). 

10.  (r^+25  +  3c+56— 4<^)(6e-5tZ-4c~36  +  2a). 

11.  {a''  +  b'  +  c''){e'-d'-c').  12.     (3^-7c^)^ 

13.  ex/(rr-36)  +  ^v/(^+3e). 

14.  e-{V{e+l)+2}+{e-Ve)V{e-'4:). 

If  a  =  5,  5  =  3,  c  =  1,  show  that  the  numerical  values  are 
equal. 

15.  Of  a'-b'  and  (a-b)  {a'+ab  +  b'). 

16.  Of  b'-c'  and  (b  +  c)  {b-c)  {b'  +  c'). 

17.  Of  a'  +  a'b'+b'  and  (a^+a5  +  ^'^)  (a'-^&4-^>'). 

18.  Of  b'+4.c'  and  {Z>^+2(^  +  ^)4{^'-2(5-c)4. 

lY.  Change  of  the  Order  of  Terms — Like  Terms. 

32.  The  terms  in  an  expression  which  are  preceded  by  the 
sign  +  are  called  positive  terms,  and  the  terms  which  are 
preceded  by  the  sign  —  are  called  7iegative  terms. 

33.  We  must  now  extend  the  meaning  and  use  of  the 
sign  —  beyond  the  strict  application  of  ordinary  arith- 
metical notions.  If  in  the  expression  a  —  b  -\-c,  «  =4,  5=7, 
and  c  =  8,  then  by  our  first  definition  of  the  sign  —  we 
should  have  to  subtract  7  from  4,  which  is  impossible.  In 
this  case  we  subtract  the  4  from  the  7  and  write  the  remain- 
der with  the  sign  -.     Thus  -7 +  4= -(7 -4)  =  -3.     Then 

Positive  and  Negative  Terms.    Algebraic  meaning  and  use  of  the  Sign  — . 


CHANGE    OF    THE    ORDER    OF    TERMS — LIKE    TERMS.      21 

we  consider  —3  +  8  to  be  the  same  as  8  —  3  =  5,  and  5  is  the 
numerical  value  of  the  expression  a—b-{-c=7—4z-\-8, 

34.  It  is  then  indifferent  in  what  order  the  terms  of  an 
Algebraic  expression  be  written.  This  is  clear  from  the  com- 
mon notions  of  Arithmetic,  and  from  the  convention  that 
—  5  +  ^  is  the  same  as  a  —  b.     Hence,  if  a  term  is  preceded 

'  hy  no  sign,  the  sign  +  is  to  be  under  stood,  and  such  a  term  is 
counted  loith  the  positive  terms,  ^ 

Thus,  7+8-2-3  =  8  +  7-2-3=:~3+8-2+7,  &c. 

a-]rb—c=b-\-a—c=b—c-\-a=—c-\-b-\'a,  &c. 

35.  Terms  are  said  to  be  like  when  they  do  not  differ  at  all 
or  differ  only  in  their  numerical  coefficients ;  otherwise  they 
are  said  to  be  unlike.  Thus  a,  4a,  and  7«^  are  like  terms; 
a^b  c,  5  a^b  c,  and  7  a^b  c  are  like  terms  ;  ba^,  6a  b,  and  6b^ 
are  unlike  terms ;  4a*  and  b  c  are  unlike  terms. 

36.  An  expression  which  contains  like  terms  may  be  sim- 
plified.    For  example,  consider  the  expression 

6a—a+3b  +  6c—b  +  Sc—2a. 

This   expression,  by  Art.   34,  is   equivalent  to 

6a—a—2a+db—b  +  6c^dc. 

JSTow  6a— a— 2a— da.  For  a  from  6a  leaves  5a;  and  taking 
2a  from  6a  we  have  da  left.  Similarly  db  —  b  =  2b ;  and 
6c-\-  dc  =  Sc.    Thus  the  expression  may  be  put  in  the  form 

3a  +  25  +  8c, 

Again,  consider  the  expression  a— 3^— 4^.  This  is  equal 
to  a  —  7^.  For  if  we  have  first  to  subtract  35  from  any 
number  a  and  then  to  subtract  Ab  from  the  remainder,  we 
shall  obtain  the  required  result  in  one  operation  by  subtract- 

riie  Terms,  in  what  order  to  be  written.  Like  and  unlike  terms.  Simplification 
of  expressions  containing  like  terms. 


^2  ELEMENTARY    ALGEBRA. 

ing  ^ib  from  a;  this  follows  from  the  common  notions  of 
Arithmetic.     Thus 

«  -  35  -  4^  =  a  -  75. 

37.  The  statement  —  35  —  45  =  —  75  is  explained  thns : 
if  in  the  course  of  an  Algebraic  operation  we  have  to  sub- 
tract 35  from  a  number  and  then  to  subtract  45  from  the 
remainder,  we  may  subtract  75  at  once  instead.  It  will  be 
seen  that  by  an  easy  extension  of  this,  the  expression  —  75 
has  a  meaning  when  standing  by  itself. 

38.  It  may  be  noticed  (as  we  have  proved  in  Arithmetic) 
that  it  is  immaterial  in  what  order  the  factors  of  a  quantity 
are  arranged.  Thus  7X5  is  the  same  as  5x7;  2x6x9  the 
same  as  6X2X9  or  9x2x6,  &c.;  abc  the  same  as  cab  or 
cba,  &c. ;  6  x^a  and  a  x^  are  like  terms.  It  is  usual,  however, 
to  arrange  literal  factors  and  terms  as  much  as  possible  in 
the  order  of  the  letters  of  the  alphabet. 

39.  The  simplifying  of  expressions  by  collecting  like  terms 
is  the  essential  part  of  the  processes  of  Addition  and  Sub- 
traction in  Algebra. 

Ex.  1.  Group  together  like  quantities,  with  their  proper 
signs,  from  5a  —  35,  4a  +  75,  and  —  8a  —  55. 


Ans.  +  5a 
+  4a 
~8a 


—35  Here  the  quantities  in  each  column 
+  75  are  lilce,  but  the  two  columns  are 
—55        unlike. 


Ex.  2.  Group  together  like  quantities,  with  their  proper 
signs,  from 

a'+  3a'5  +  3a5^+  2a'+  25^4-  5a5^-  Sac''-  a^b  -  b\ 

-Sac' 


Ans.     +  a'  I  -f-3a'5 

+  2a^  I  -  a^b 


+  5a5' 


+25' 
—  b^ 


Ex.  3.  Group  together  lilce  quantities,  with  their  proper 
signs,  from   2a  -  35  +  75c  +  5'c  -  5a5c  +  2.Ty  -3:c'  +  55"  + 

Order  of  arraK<rement  cf  Factors. 


CIIAKGE    OF    THE    ORDER    OF  '  TERMS. — LIKE    TERMS.     23 


Wc  -  9a  -  W  +  6^  4-  10a  -  hx'^  -  xy  ^  x^  -\-  ahc  -  2bc  +  c' 
-b-  dc\ 


Ans. 


+  2a 

-3b  -{•7bc\+  Fc 

—oabc\-\-2xy 

-^x" 

■VW 

—  9a 

-4-6^ 

-2bc  -\'Wc 

+  abc 

—  ^y 

-bx' 

-2b' 

+  10a 

-  b 

+  x" 

40.  We  shall  close  this  chapter  with  some  more  examples 
of  the  conversion  of  ordinary  language  into  Algebraic  lan- 
guage. 

Ex.  1.  A  person  makes  a  mixture  of  three  sorts  of  wine. 
The  second  costs  a  dollars  more  per  gallon  than  the  first ; 
and  the  third  b  dollars  more  per  gallon  than  the  second. 
There  are  m  gallons  of  the  first  sort,  n  of  the  second,  and 
p  of  the  third.  What  is  the  price  of  the  mixture,  x  being 
the  price  of  the  first  sort  ? 

Ans.    mx  ■{-  n  (x  -\-  a)  ■\-  ;p  {x  ■\-  a  ■\-  b), 
Ex.  2.  Express  algebraically  a  number  of  five  figures  a,  b, 
c,  d,  e,  taken  in  their  order  from  left  to  right  in  the  decimal 
system. 

Ans.  lO'Xa  +  10'X^  +  lO'Xc  +  lO^+e. 
Ex.  3.  Three  fountains  run  successively  into  a  reservoir, 
the  first  during  a  hours,  the  second  during  b  hours,  the 
third  during  c  hours ;  the  second  fountain  supplies  m  gallons 
per  hour  more  than  the  first,  the  third  n  gallons  more  than 
the  second  :  how  many  gallons  of  water  did  the  three  foun- 
tains yield,  x  being  the  number  of  gallons  per  hour  which 
the  first  yields  ? 

Ans.  ax  +  b  (x  +  m)  -\-  c{x  -\-  m  +  n). 
Ex.  4.  A  merchant  sells  a  certain  number  of  yards  of  cloth 
for  a  dollars  per  yard.  A  second  merchant  sells  b  more 
yards  of  the  same  cloth  at  c  dollars  more  per  yard.  If  a;  is 
the  number  of  yards  sold  by  the  first,  express  the  difference 
in  the  amounts  received  by  the  two. 

Ans.     {x-\-b){a-\-c)  —  ax. 


24  ELEMENTARY    ALGEBRA. 

Ex.  5.  A  mixture  is  made  of  4  substances,  A,  B,  C,  D.    It 

is  composed  of  a  gallons  of  A,  costing  m  dollars  per  gallon, 

of  h  gallons  of  B  at  n  dollars  a  gallon,  of  c  gallons  of  C  at 

2)  dollars  a  gallon,  and  of  d  gallons  of  D  at  g  dollars  a  gal 

Ion.     Express  tlie  price  of  a  gallon  of  the  mixture. 

,  am  +  bn  -\-  cp  +  da 

Ans.    • ;-, — 7-—. 

a  +  0  -\-  c  +  a 

Ex.  6.  Three  laborers  paid  at  the  same  rate  worked,  the 

first,  m  days,  the  second  n  days,  and  the  third  q  days.     They 

received  altogether  a  dollars.    Express  the  daily  wages  of 

each. 

Ans.    — ■ -— . 

m+n+q 

Ex.  7.  A  sum  a  produced  in  b  years  c  dollars  at  simple  in- 
terest.    Express  the  rate  per  cent. 

100c 

Ans.    -7 — . 
ba 

Ex.  8.  A  sum  a  placed  at  simple  interest  amounts  in  n 

years  to  b  dollars.    What  is  the  rate  per  cent.  ? 

Ans.     m^-^. 
an 

V.    Addition. 

41.  It  is  conyenient   to   make  three   cases  in   Addition, 
namely :  I.  When  the  terms  are  all  like  terms  and  have  the 
same  sign.     II.  When  the  terms  are  all  like  terms,  but  have 
not  all  the  same  sign.    III.  When  the  quantities  to  be  added 
consist  of  both  like  and  unlike  terms. 
■  42.     I.  To  add  like  terms  which  have  the  same  sign. 
Add  the  numerical  coefficients,  prefix  tJie  common  sign  to 
the  sum,  and  annex  the  commo7i  literal  factors. 
For  example : 

6a  +  3a  +  7a-{-6a  =  21a. 
-  3b'c  -  bb'c  -10b' c  =  -  ISb'c. 

Three  cases  in  Addition,    Rule  for  Case  I. 


ADDITION".  25 

43.  II.  To  add  like  terms  which  haye  not  all  the  same 
sign.  Add  separately  the  positive  numei^ical  coefficients,  and 
the  negative  numericcd  coefficients  ;  take  the  difference  of  these 
tivo  sums,  prefix  the  sign  of  the  greater  to  this  difference,  and 
annex  the  common  liter cd  factors. 

For  example : 

7rr  -  3^f  +  11«'  +a'  -  oa'  -  2a'  =  19a'  -  10a'  =9a'. 
%l)0  _  ^ic  -  Zlc  +  ^hc  +  hhc  -  Q>hc  =  llhc—l^hc  =  -  6bc. 

44.  III.  To  add  expressions  which  consist  of  both  like 
and  nnlike  terms.  Add  together  the  like  terms  by  the  rule  in 
Case  II,  Affix  to  the  sums  thus  obtained  the  unlike  terms^ 
each  preceded  by  its  proper  sign, 

For  example :  add  together 

4a-f  5^-7c  +  3f/,  3a-&  +  2c+5^,  ^a-^b-c-d, 

and  —a  +  'db-\-4:C—M^-e, 

It  is  conyenient  to  arrange  the  terms  in  columns^  so  that 
like   terms    shall    stand    in    the   same    column;    thus  we 

haye 

4a+5^-7c+3f? 

3a-  b-\-2c-VM 
9a— 2b—  c—  d 
-a-^Zb^4.c-M-\-e 


lDa-\-bb—2c^^d-\-e. 


Here  the  terms  4a,  3a,  9a,  and  —a  are  all  like  terms;  the 
sum  of  the  positiye  coefficients  is  16  ;  there  is  one  term  with. 
a  negative  coefficient,  namely  —a,  of  which  i\\Q  coefficient 
is  1.  The  difference  of  16  and  1  is  15 ;  so  that  we  obtain 
-f-  15a  from  these  like  terms :  the  sign  +  may,  however,  be 
omitted.  Similarly  we  have  bb  —  b  —  2b  -\-  ^b  =  6b.  And 
so  on. 

Rule  for  Case  II ;  for  Case  III. 

2 


26 


ELEMENTARY    ALGEBRA. 


45.  In  the  following  examples  the  terms  are  arranged  suit- 
ably in  columns. 


4:x'  +  '7x''+  x-9 
—  2x'+  x'-  dx  +  S 
-3x'-  a;'  +  10.T-l 

dx'-     x-1 


a'-h  ab+  b'-c 
a^-'Zab-W 


In  the  first  example,  we  have  in  the  first  column 
x^  +  4zX^ —2x^ —Zx^ ,  that  is,  bx^—bx^,  that  is,  nothing;  this  is 
usually  expressed  by  saying  the  terms  ivhicii  involve  x^  cancel 
each  other. 

Similarly,  in  the  second  example,  the  terms  which  involve 
ab  cancel  each  other;  and  so  also  do  the  terms  which  in- 
volve z>^ 

Ex.     Add  together  a  +  ^b  —  c,  a—6e-h2c,  and  x  +  y  +  3e. 


Here  a  and  a  are  UJce, 

—  6e  and  +3e 

—  c  and  +2c 

the  rest  are  imlike. 


a  +  2b—  c 
a—5e-\-2c 
3e  +  x  +  y 

Sum  =:  2a  +  2b  +  c—2e  +  x-\-i/, 


Ex.     Add  together   Za^  —  bc,  2b^—ac,  4cG^  —  ab,  and   a^  +  b'^ 
^c\ 


Here  3a^  and  a^  are  lihe, 
2^^  and  ^-b^ 
4c^  and  —  c' 
the  rest  are  unlike. 


3a^-bc 

2b' -ac 

^&-ab 
a'+  y-  c' 


Sum  =  4ca^  +  3b'  -{-3c'-ab-ac  -  be. 


ADDITIOX.  27 


Ex.    Add  together  xy—1,  vc'-f  2,  and  ^''  +  3. 

Here  the  terms   are  all  x^  +  2 

icnlilce,  except  —1,  +2,  and 
+  3. 


^^  +  3 


Sum  =  x^  +  xi/-^y^  +  4:. 


46.  The  Kules  aboye  given  for  the  Addition  of  like  and 
imliJce  algebraical  quantities  are  in  no  wise  different  from 
those  employed  in  Arithmetic.  For,  suppose  we  have  to  add 
together  3  hundreds  and  4  hundreds,  we  combine  these  lihe 
quantities  by  taking  the  sum  of  the  coefficients  3  and  4,  so  as 
to  make  7  hundreds.  But  if  we  have  to  add  together  3 
hundreds,  5  tens,  and  6  units,  these,  being  unlike  quantities^ 
cannot  be  added  in  the  same  sense,  but  are  merely  collected 
together  in  one  line,  3  hundreds  +5  tens  +6  units,  which, 
for  convenience,  is  written  shortly  356.  It  will  be  observed, 
however,  that  algebraical  Addition  involves  the  processes 
both  of  arithmetical  Addition  and  Subtraction. 

Examples — 5. 

Add  together 
1.  a  +  5  and  a^-h,  ^,  a^h  and  a—h, 

3.  G^— 6  and  a— ^.  4.  «  — S  +  c  and  a4-^— f-. 

5.  a—h-\-c^VL^a-\-h-^c,      6.  1  — 2m  +  37^  and3m— 2;i  +  l. 
7.  5m +  3  and  %m  —4.         8.  Zxy—%x  and  xy^Qx, 
9.  4;9-25'  +  l  and  7-3^?-!-^. 

10.  5aZ>  — 2^c  and  ah -{-be, 

11.  3a~2^,  4a-55,  7a-115,  a+95. 

12.  4a;' -3?/',  2x^-by\  ~x^-Vy\  -22:'-^4y^ 


28  ELEMEXTARY    ALGEBRA. 

13.  5a-i-3I)  +  Cy  da  +  3b  +  3c,  a  +  db  +  oc. 

14.  3X  +  22J-Z,  2x-2ij  +  2z,  -x-\-2y-^3z. 

15.  "/a-U  +  c,  6a  +  3b-6c,  -12a  +  4.c. 

16.  x—4:a  +  b,  3x  +  2I),  a—x—hl, 

17.  a-\-l—c,  h  +  c—ay  c^-a—l,  a^l—c. 

18.  a4-2Z>  +  3c,  2a— I— 2c,  l—a—c,  c—a—K 

19.  a— 2J  +  3c— M  ?>l}—4.c-\-Dd—2a,  5c-6^+3a— 45, 

20.  x'-^x''-\-bx-Z,2x'-W-li:X-\-^,  -x'-^-^x^'-Vx-^S. 

21.  a;'-2:^'  +  3z',ix;'  +  a;'^  +  :?;,  4:^;'  +  5:c^  2a;'^  +  3:?;-4, 
-32:^ -2.-^-5. 

22.  a'-'^cn-{-'daV-d\  2a'  +  6a'I)-6ad'-7b\ 
a'-ab'  +  2I?\ 

23.  x'--2ax'  +  a''x  +  a%  x'  +  dax\  2a^-ax^-2x\ 

24.  2a'b—Zax^ ^2(j^x,  \2db-\-V)adi^  —  ^(^Xy 
—  Sal?  +  ax^—5a^x, 

25.  2;^  +  y*  +  ;^^  -4.x'-6z%  Sx'-lif  +  lOz',  6t/-6z\ 

26.  32;' -  4:?;?/ 4- ^'  +  2:^;  + 3?/ -7,  22;'-4?/'-h32;-5y  +  8, 
lOa;?^'  4-  8^'  +  9y,  5a;'  -  62;?/  +  3/  +  72^-7^  +  11. 

VI.    Subtraction. 

47.  Suppose  we  have  to  take  5  +  3  from  14.  The  result  is 
the  same  as  if  we  first  take  5  from  14,  and  3  from  the  re- 
mainder.   This  result  is  denoted  by  14  —  5  —  3. 


SUBTRACTION^.  29 

That  is,  14  -(5  +  3)  =  14  —  5  —3.  The  brackets  mean- 
in  2:  that  the  whole  of  the  5  +  3  is  to  be  taken  from  14. 

In  like  manner,  suppose  we  have  to  take  'b-{-c-{-d  from  a. 
The  result  is  the  same  as  if  we  first  take  h  from  a^  and  then 
take  c  from  the  remainder,  and  then  d  from  that  remainder; 
that  is,  the  result  is  denoted  by 

a—h—c—d. 

Thus 

a~i^^c^-d)=a—'b—c—d. 

We  see  in  these  cases  the  positive  terms  of  the  expression 
to  be  subtracted  have  all  been  changed  to  negative  terms  in 
the  result. 

48.  l^ext  suppose  we  have  to  take  5  —  3  from  14.  If  we 
take  5  from  14  we  get  14—5;  but  we  have  taken  too  much 
from  14,  for  we  had  to  take,  not  5,  but  5  diminished  by  3. 
Hence  we  must  increase  the  result  by  3.  The  result  is  then 
denoted  by  14—5  +  3. 

Thus 

14-(5-3)  =  14-5  +  3. 

In  like  manner,  suppose  we  have  to  take  h—c  from  a.  If 
we  take  l  from  a,  we  obtain  a—h'^  but  we  have  thus  taken 
too  much  from  a,  for  we  had  to  take,  not  Z>,  but  Z>  diminished 
by  c.  Hence  we  must  increase  the  result  by  c.  Thus  we 
obtain 

a— 1)^-0, 
That  is 

a—{h—c)—a-''b^-c. 
By  the  same  reasoning 

a—{h—c—d)=a—l)-\-c-\-d. 

Here  the  positive  term  of  the  expression  to  be  subtracted 
is  negative  in  the  result,  and  the  negative  terms  are  posi- 
tive. 


80  ELEMEKTAKY    ALGEBRA. 

49.  Hence,  we  have  tlie  following  rule  for  Subtraction  : 
Change  the  signs  of  all  the  terms  in  the  ex^jression  to  be 

stihtracted,  and  then  proceed  as  in  Addition, 

For  example:  from  4:X—?>y -\-2z  subtract  2x—y-\-z, 
Change  the  signs  of  all  the  terms  to  be  subtracted ;  thus 

we  obtain  —^x  +  y—z'y   then  collect  the  terms,  and  simplify 

as  in  Addition.     Thus 

4:X—'dy  +  2z—'dx  +  y—z=x—'^y-\-z. 

From   ^x'  +  bx'-Qx^-'lx  +  b,   take   2.x*-2aj'  +  5cc'-6a;-7. 
Change  the  signs  of  all  the  terms  to  be  subtracted,  and 
proceed  as  in  Addition.    We  thus  have 

3aj'  +  5cc'-6x'-7cc  +  5 
-2a3^  +  2aj'-5x'  +  6aj-f7 


«j'  +  7i«'~llaj'-cc  +  12. 


The  beginner  will  find  it  best  at  first  to  go  through  the 
process  fully  as  above ;  but  he  will  soon  learn  to  put  down 
the  result  without  actually  changing  all  the  signs,  but 
merely  doing  it  mentally. 

50.  We  often  have  a  single  negative  term  to  be  subtracted 
from  another  term  or  expression.  Thus,  from  a  subtract 
—  c.  Here  we  can  reason  thus:  Since  a=a  +  'b—'b,  if  we  sub- 
tract —  h  from  a,  the  result  is  a  +  h,  the  same  as  if  we  add  +  h 
to  it.  Or  we  can  apply  our  rule,  at  once  considering  the 
result  to  have  a  meaning  in  connection  with  some  other 
parts  of  an  algebraical  operation. 

Examples. 

1.    From  ^a  2.    From  la  3.    From  a 

take    a  take  6a  take  a 

Ans.     2a  a  0 

Rule  for  Subtraction. 


SUBTRACTION. 

31 

4. 

From 

3a 

5. 

From      7a 

6. 

From      a 

take 

—a 

take  —6a 

take  —a 

Ans. 

4a 

13a 

2a 

7. 

From 

~3a 

8. 

From  —7a 

9. 

From  —a 

take 

Ans. 

a 

~4.a 

• 

take      6a 

take     a 

-13a 

-2a 

10. 

From 

—3a 

11. 

From  —7a 

12. 

From  —a 

take 
Ans. 

—  a 

take  —6a 
—  a 

take  —a 

-2a 

0 

13. 

From 

a-hb 

14. 

From  a—b 

15 

.     From  y  +  ax 

take 
Ans. 

a-b 
2b 

take  a  +  b 
-2b 

take  y—ax 

2ax 

16. 

From  3a-U  +  6c 

17. 

From 

7a- 

-2b-\-4:C-2 

take 
Ans. 

a— 2Z>  +  9c 

take 

6a' 

-6b-\-4.c-l 

2a-2b- 

-3g 

a  +  ^b-1 

18.    From  2a— 6a^—  ac  +  5       19.     From  3cc?/— cc'—  y^'  +  a 
take  5a~8aJ— 2ac— 1  take  2xy  +  x^-\-2y'^  —  b 


An  s.      —  3  a  +  2a^  +  ac  +  6 


xy—2x''  —  3y''  +  a  +  b 


20.     From     a^  +  2aJ-36'' 

take  2a'^  —  Dab—7c^ 

Ans.      -a'  +  7a2>+4c' 


21.     From  bx"^-  xy+  y^ 

take  —  a3^  +  4i^^  +  3^^ 

6x'-6xy-2y^ 


32  ELEMENTARY     ALUt:i3EA. 

22.     'FiomSa'+x'-bb'-Dc''        23.     From  ^'- 3a;' +  6:^;- 10 
take  x'-{-2F-6c''  tske  x'-4.x'-^8x-  9 


Ans.      8a' -n'  x^-%x-\ 


24.    From  a-\-\h-\-\  25.    From  \x^—\xij^\if 

take  2^  +  ^  +  2  '  ^^^^  '~3^'~i^^~2^' 


111 

Ans.     -ir^-17^4-^  x^—xy-\rm^ 

z       z      z 

Examples — 6. 

1.  From  7a +  14^  subtract  4a  +  10Z>. 

2.  From  6a--2J— c  subtract  2a— 2^— 3(?. 

3.  From  3a— 2Z>  +  3c  subtract  ^a—lh—c—d. 

4.  From  "Ix^—^x—l  subtract  bx'—Qxi-3. 

5.  From  4.x' -2>x' -2x^-^1  x-^^ 

subtract  x' — 2x^ — 2x^  +  7x—9, 

6.  From  2x^  —  2ax+3a^  subtract  x^—ax  +  a^. 

7.  From  x'—3xy—y''-]-yz—2z' 

subtract  x^ + 2xy  +  bxz — 3  ?/'  —  2^;'. 

8.  From  hx''  +  Qxy-12xz-4.y''-^2jz-bz^ 

subtract  2a;' -  7a;?/  +  4.T;a;- 3?/'  +  ^yz— ^z\ 

9.  Froma'-3a'Z^  +  3aZ>'-Z^'' 

subtract  -a' ^-^cn-^alf +  1)\ 

10.  From  7a;' -  2a;' +  2a; +2  subtract  4a;' -2a;' -2a' -14, 
and  from  the  remainder  subtract  2a;'  — 8a;'-f-4:c+16. 


VII.    Brackets. 

51.  On  account  of  the  extensive  use  of  brackets  in  the 
algebraic  language,  it  is  necessary  that  the  student  should 
observe  very  carefully  the  rules  respecting  them. 

Since  the  sign  +  or  —  preceding  a  bracket  means  that 
the  whole  included  quantity  is  to  be  added  or  subtracted,  if 
we  wish  to  remove  the  bracket,  we  must  actually  perform 
the  operation  indicated  by  means  of  it;  i.  e.,  we  must  add  or 
subtract  the  quantity  in  question.  ISTow  when  a  quantity  is 
added,  the  signs  of  its  terms  are  not  altered ;  but  when  it  is 
subtracted,  the  signs  of  its  terms  are  changed. 

Hence, 

When  an  expression  is  ivitliin  a  pair  of  brackets  preceded 
ly  the  sign  -\-,  the  hrachets  may  he  removed,  the  signs  of  the 
included  terms  'being  unchanged. 

When  an  expression  is  ivithin  a  pair  of  hracJcets  preceded 
ly  the  sign  — ,  the  hraclcets  may  le  removed  if  the  sign  of 
every  term  luithin  the  hraclcets  he  changed. 
Thus,  for  example : 

a—'b-\-{c'-d-\-e)=^a—'b-\-c—d-\-e 
a—I?  —  {c—d-\-e)  =  a—h—c-\-d—e, 

Kemember,  that  if  the  first  term  within  the  brackets  has 
no  sign,  the  +  sign  is  understood  before  it. 

^2,  In  particular,  the  student  must  notice  such  statements 
as  i\\Q  following: 

These  are  immediate  consequences  of  what  we  have  said 
of  the  addition  and  subtraction  of  single  terms. 

Eules  for  removing  Brackets. 

2^ 


34  ELEMENTARY    ALGEBRA. 

53.  Expressions  may  occur  with  more  than  one  pair  of 
brackets ;  these  may  be  remoyed  in  succession  by  the  pre- 
ceding rules,  beginning  with  the  inside  pair. 

Thus,  for  example : 

a+{l)-{-{c—d)}=a+{b  +  c—d]=a  +  'b-\-c—d, 

a-v{b—{c—d)]=a^-{'b—c-^d]—a^-'b—c-\-d, 

a— '  {!)-{- {c—d)}=a—{'b^- c—d]-=ia—'b  —  c-{- d, 

a—  {l)—{c—d)}  ^a—  {l)—c-[-  d)  =a—d  -{-  c—d. 

Similarly, 

a—[b—{c—(d—e)}'\=a—[h—{c—d+e]'\ 

=a—[b—c-\-d—'e]=a—I)  +  c—d  +  e, 

It  will  be  seen  in  these  examples  that,  to  prevent  confu- 
sion between  various  pairs  of  brackets,  we  use  brackets  of 
different  shapes ;  we  might  distinguish  by  using  brackets  of 
the  same  shape  but  of  different  sizes. 

A  vinculum  is  equivalent  to  a  bracket ;  see  Art.  30.  Thus, 
for  example : 

a-\h-{c-{d-^)]l^=a-\l-{c-{d-e-^f)]\ 

=a-\h—{G-d-\-e-f]^=a-\l-c  +  d-e+fl^ 

^a—l  +  c—d-^-e—f. 

54.  The  beginner  is  recommended  to  remove  brackets  in 
the  order  shown  in  the  preceding  article,  {i.  e.)  by  removing 
first  the  innermost  pair,  next  the  innermost  pair  of  those 
which  remain,  and  so  on.  We  may,  however,  vary  the 
order,  by  removing  first  the  outermost  pair,  next  the  outer- 
most pair  of  those  which  remain,  and  so  on. 

Thus,  for  example : 

a-\'{h-\-{c-d)}=a-hh-\-{c-d)=a  +  b  +  c—dy 


BEACKETS.  35 

a+  [b— {c— d)}  =  a  +  b— {c— d)  -—a  +  I)—c  +  d, 
a—{b  +  {c  —  d)}=a—I)  —  {c—d)=a  —  I)  —  c  +  d, 
a—{b  —  {c—d)}=a—b-{-{c—d)  =  a--b-\-  c—d. 
Also, 
a—[b-{c-{d—e)}]^a-b+{c—{d-e)} 

=a—b  +  c—{d—e)~a—b-{-c—d+e, 

55.  It  is  often  convenient  to  take  up  in  brackets  any  given 
terms  of  an  expression.  Tlie  rules  for  thus  introducing 
brackets  follow  immediately  from  those  of  removing 
brackets. 

Any  oiiwiber  of  terms  'i?i  an  expression  may  be  put  loitJmi 
a  pair  of  braclcets,  and  the  sign  +  pUtced  before  the  brachet, 
the  signs  uf  the  terms  being  unchanged. 

Any  number  of  terms  in  an  expression  may  be  put  icithm 
a  pair  of  braclcets,  and  the  sign  —  placed  before  the  brachet, 
'provided  the  sign  of  every  term  put  loithin  the  brackets  be 
changed. 

In  applying  this  rule,  we  shall  for  convenience  take  the 
sign  of  whatever  term  we  choose  to  set  as  first  term  within 
the  brackets,  as  the  sign  to  be  placed  before  the  bracket. 

Thus  -\-a—b  —  c,  collected  in  a  bracket  with  -\-a  SiS  first 
term,  will  be  +{a—b—c);  but,  with  —b  as  first  term,  it 
will  be  —{b  —  a  +  c),  and  with  —c  as  first  term,  it  will  be 
—  {c'-a  +  b);  and  now,  if  we  resolve  again  these  last  two 
brackets,  the  sign  (  — ),  preceding  each  of  them,  will  correct 
the  changes  we  have  made,  and  the  quantities  will  be  repro- 
duced, as  at  first,  —b  +  a—c,  —c-i-a—b. 

So  also  we  might  use  an  inner  bracket,  and  write  the 
quantity  -\r  {{a~b)—c},  or  +  {a—{b  +  c)},  or  —{{b  —  a)  +  c}, 
or  —{b—{a—c)},  &c. 

Rules  for  intv^d'^clng  Bracketg. 


'56  elementary  algebra. 

Examples — 7. 
Eeduce  to  their  simplest  forms : 

1.  {a-x)-{2x-a)~{2-2a)  +  {3~2x)-{l-x). 

2.  {a'  -  2a' c  +  3ac')  -  {a'o  -  2a'  +  2ac')  +  {a'  -  ac'  -  a'c). 

3.  {2x'~2jf-z')-{3if  +  2x'-z')-{3z'-2tf-x'). 

4.  {x'  +  ax'  +  a'x)-{i/-I?y'-hh'i/)  +  {z'  +  cz'  +  c'z) 

-  i^'-y"  +-^")  +  {cix'  +  hf  +  cz')  -  {a'x-Fij  +  c'z). 

5.  a"-  {W-c')  -  {b'-  {c'-a')]  +  [c'-  ifi'-a')}. 

6.  {2a'-{dad-F)}-{a'-{4.ad  +  W)]-\-{2h'-{a'-al))]. 

7.  {x'^f-{dx\j-^3xif)}-{{x'-3xhj)  -  Cdxif-if)}. 

8.  {'^x-{3y-z)}-{y-\-{2x-z)}-\-{3z-{x-2y)] 

^{2x-{y-z)}. 

9.  l_{l-(l-4r^)}  +  {2a;^(3-5:r)}-{2~(-44-5a;)}. 

10.  {2a-{3I)-\-c-2cl)]-{{2a-3h)-{-{c-2d)] 

>    '\-{2a-{3h-\-c)-2d)}-{{2a-3'b-\-c)-2d}. 

Express  by  brackets,  taking  the  terms  (i)  tivo  together,  (ii) 
three  together : — 

11.  a-2l-{-3c—d-\-2e—f.  12.  —2'b-\-3c—d-^2e—f-\-a. 
13.  3c-d^2e-f-\-a-21).  14.  -d-^2e-f-\-a-2'b-Y3c, 
15.    2e-f-]-a-21)-\-3c-d.       16.     ^f^a-21)^-3c-d-\-2e 

56.  In  Addition  and  Subtraction  we  have  spoken  hith- 
erto only  of  numerical  coefficients;  but  as  any  one  of  the 
factors  of  which  a  term  is  composed  may  be  considered  a 
coefficient,  we  often  have  to  apply  the  rules  to  these  literal 

Rules  for  Brackets,  as  applied  to  literal  coefficients. 


BRACKETS.  37 

coefficients.  Thus,  when  any  terms  of  a  quantity  contain 
some  common  factor,  brackets  are  often  employed  to  collect 
tlie  other  factors  considered  as  its  literal  coefficients  into  one 
expression,  which  is  set  before  or  after  the  common  factor. 

Thus,  just  as  dx-{-2x—x=4:X,  that  is,  ={3-\-2—l)x, 

so,  likewise,  ax-]-dx—x={a+I)—l)Xf 

2a-4.ax  +  6ay=2a{l-2x-\-3i/), 

{a+2h)x'-{2d-c)x'={{a+2I))-{2d^c)}x'={a+c)x\ 

Ex.  1.  Add  {a-2jo)x''+{2c-3r)x 

{22J-{-a)x^  —x 

'—{p—a)x^    —{c^l)x 

^x^  —{c—2r)x 

Ans.   {^a—p—Vjx^  —rx 

Ex.  2.  From        ax'        -M  +x 

take     —px'         ^qx^  -\-rx 

Ans.    (a-\-p))x^  —  (b—q)x^  +  {l—7')x 


Examples— 8. 

1.  Collect  coefficients  in  ax^  —  lx'^  —  cX'—lx^-\-cx^~-clx-\-cx^ 

—dx^—ex, 

2.  Add  together  ax—ly,  ^-\-y,  and  {a'~l)x—{l)-\-l)y. 
Jl.    Add  together  {a-\-c)x^—3{a—'b)xy-\-{ib—c)y'',  and 

{l)-c)x'-\-2{a^l)xy  +  (a-'b)y\ 
4.    Add  together  {a  +  h)  x  +  {I)  +  c)y  and  {a—'b)x—{h~c)y, 
and  subtract  the  latter  from  the  former. 


38  ELEMEKTAEY    ALGEBRA. 

5.  Add  together  (i)  the  first  two,  (ii)  the  last  two,  and 

(iii)  all  four  together,  of  '^{a+'b)x-^^{l)  +  c)y, 
-'3{a-I))x  +  2{a-c)t/,  —(2b-^c)x+{a-2J))y,  and 
{a-2l)x-{h  +  %c)ij. 

6.  In  (5)   (i)  subtract  the  second  quantity  from  the  first, 

and  (ii)  the  fourth  from  the  third,  and  (iii)  add  the 
two  results  together. 

YIII.    Multiplication. 

57.  It  is  convenient  to  make  four  cases  in  Multiplication. 
I.  To  multiply  one  positive  single  term  by  another ;  II.  To 
multiply  a  quantity  consisting  of  two  or  more  terms  by  a 
positive  single  term ;  III.  To  multiply  one  quantity  by  an- 
other when  both  consist  of  two  or  more  terms ;  IV.  To  mul- 
tiply one  negative  single  term  by  another,  or  by  a  positive 
single  term. 

58.  I.  Suppose  we  have  to  multiply  da  by  4Z>.  The  prod- 
uct may  be  written  thus,  ZaX^l)]  or,  since  the  product  of 
any  number  of  factors  is  the  same  in  whatever  order  the  fac- 
tors may  be  taken,  we  may  write  it  dX^XaXh',  and  it  is 
therefore  equal  to  V^ah,  Here  we  multi2:)ly  together  the  nu- 
merical coefficients  and  put  the  literal  factors  after  this 
product. 

Thus,  for  example : 

laXdhc^^lalc. 
Similarly, 

4.aXhl)X'dc=ma'bc, 

59.  Poivers  of  the  saine  numler  are  multiplied  together  hy 
adding  the  exponents. 

Four  Cases  in  Multiplicatiou.  Rule  for  Case  I.  Powers  of  the  same  luim 
ber,  how  multiplied? 


MULTIPLICATION.  39 

Thus,  a'^Xa^—a'']  for  a'—aa,  and  a^~aaa,  ^'.a'Xa^ 
=aaX  ciaa = aaaaa,  or  a^ 

In  the  same  manner  it  maybe  shown  that  a^Xa^"=a'"; 
and  so  on  for  other  powers,  always  taking  the  sum  of  the 
exponents.     To  prove  this  generally,  viz.,  that 

oJ^XdP'^dr-^'^,  whatever  positive  whole-numbers  m  and  n 
may  stand  for,  we  have,  by  definition, 

oJ^=za.aM,  &c.  to  m  factors, 

and  a^'^aM.a.  &c.  to  n  factors, 

:,  a'''Xa''=a.a.a,  &c.  to  m  factors  x  a.a,  &c. to  n  factors, 

—a, a, a.  &c.  to  7n  +  n  factors, 

—  a!^^'^,  by  definition. 

The  reasoning  and  the  rule  are  the  same,  if  for  a  we  wnte 
a-^l),  or  a-\-'b^c,  or  any  other  quantity ;  that  is,  the  poioers 
of  such  quantities  are  multiplied  together  by  adding  the 
exponents  of  the  powers  together.  Thus  the  2d  power  of 
a^h  multiplied  by  the  3d  power  of  the  same  quantity  will 
produce  the  5th  poAver  of  that  quantity. 

Ex.  1.     2vT'x3.^•'=2x3X.^•V=:6.^^ 

Ex.  2.     "^ax  X  ^axy  ==  7  X  2  X  aaxxy — UaVy. 

Ex.  3.     ha%Xa'bc  =  oa\iUc=^^a''h''c, 

Ex.  4.     ?>x'y'z'  X  4.xYz = 3  X  4  X  xVyYz'z = 12xyz\ 

Ex.  5.     7n7ix^yXpy=ninpx^yy—mnpx^y^, 

Ex.  6.     2r/"'  X  3a'  =  2  X  3  X  a'^a'  =  6a"^+^ 

Ex.  7.     ax'^Xix''  =  abx'''x''=aI)x'''+\ 

Ey  8.     ax'"Xbx''Xcx''=:abcx'^x''x''  =  ccI)cx''''^''+^. 

60.  11.  Suppose  we  have  to  multiply  a-f  Z>  by  3.     We  have, 
3[a  +  b)=a  +  t?-\-a  +  b+a+b=:3a-{-Sb. 


40  ELEMENTARY    ALGEBRA. 

Similarly, 

In  the  same  manner  we  have, 

3{a—b)  —  da—3b  c{a—l)  —  ca—cb 

Tims,  to  mnltiply  an  expression  consisting  of  two  or  more 
terms  by  a  single  positive  term:  Multiply  each  term  of  the 
expression  hy  the  single  term,  and  put  'before  each  product  the 
sign  of  the  term  tvhich  produced  it  j  then  collect  these  results 
to  form  the  complete  product. 


Ex. 

1. 

a-^-b—c  multipl] 

Led  by  2=2a  +  2J-26*. 

Ex. 

2. 

a—b-^-c 

d=ad—bd  +  cd. 

Ex. 

3. 

ax-\-by 

c—acx-\-bcy. 

Ex. 

4. 

ax-\-by-'Cz 

2p = 2apx + 2  b^jy — 2  cpz. 

Ex. 

5. 

2a  +  db—4:C      .. 

2x=4:ax+6bx—Scx, 

Ex. 

6. 

ax-\-by 

ax=aV-{-abxy, 

Ex. 

7. 

ax^by 

by=abxy-\-Fy^, 

Ex. 

8. 

7a;-4y+6       ... 

3:r  =  2l2;'- 12^:^  +  18:*. 

Ex. 

9. 

^X^'-\?>X-\-\     ... 

6  =  30x'-Gox+5. 

Ex. 

10. 

x^—px-\-q^ 

px  —px^  —p^x^  -\-pgx. 

Ex. 

11. 

\ah^\cd 

.       4.ab=2a'V-VGal)cd. 

61.  III.  Let  it  be  required  to  mnltiply  a-\-b  by  c-\-d\ 
this  means  that  a-\-b  \^  to  be  taken  c-\-d  times,  that  is,  g 
times  and  d  times.  Kow  a-\-b  taken  c  times  produces,  by 
rule  of  Art.  60,  ac-\-bc',  and  a-\-b  taken  d  times  produces, 

Rule  for  Case  IL 


MULTIPLICATIOIS".  41 

by  tlie  same  rule,  ad-^-hd;  .*.  a-{-b  taken  c  times  aiid  d 
times,  that  is,  c-{-d  times,  produces  ac-^-hc-^-ad-^-ld,  wliicli  is 
the  product  required. 

Or,  if  the  quantities  ho,  a-\-'b  and  c—d,  a+h  multiplied  by 
c—d  means  that  a+i  is  to  be  taken  d  times  less  than  c 
times.  Now  a+b  taken  c  times  produces  ac-\-bc;  but  this 
is  too  much  by  d  times  a+b,  that  is,  by  ad+hd;  .*.  ad+M 
is  to  be  subtracted  from  ac-\-bc.  Hence  the  product  re- 
quired is  ac+bc—ad—bd,  following  the  rule  of  Subtrac- 
tion. 

Or,  if  the  quantities  be  a—b,  and  c—d,  the  product  of 
these  is,  as  in  the  last  case,  c  times  a—b  wanting  d  times 
a—b,  that  is,  ad—bd  siMracted  from  ac—bc^  which  leaves 
ac—bc—ad-\-bd  (changing  the  signs  in  the  quantity  to  be 
subtracted,  according  to  rule). 

Collecting  these  results,  we  have, 

(a  +  b)  (c-\-d)—ac-\-bc-\-ad+ bd 

{a  +  b){c—d)  =  ac  +  bc—ad—bd 

{a—b){c—d)  =  ac—bc—ad^-bd. 

Considering  these  results,  we  see,  for  example,  that  cor- 
responding to  +  «  in  the  multiplicand,  and  +  c  in  the  multi- 
plier, there  is  a  term  -\-ac  in  the  product;  corresponding  to 
the  terms  +a  and  —d  there  is  a  term  —ad  in  the  product; 
corresponding  to  the  terms  —b  and  +c,  there  is  a  term  —bo 
in  the  product;  and  corresponding  to  the  terms  —b  and  —  ^ 
there  is  a  term  +  bd  in  the  product. 

These  observations  are  briefly  collected  in  the  following 
important  rule  in  Multiplication:  Lihe  signs  produce  +, 
and  unlike  sig7is  — .      This  rule  is  called  the  Enle  of  Signs. 

62.     IV.  Let  it  be  proposed  to  multiply  2a  by  —4zb,  or 

The  Rule  of  Si^us. 


42  ELEMENTARY    ALGEBRA. 

—  4:C  by  da,  or  -4c  by  -4^.     We  apply  the  Eule  of  Signs, 
aboye  establislied,  to  these  single  terms.     Thus,  we  have, 
2aX-^h='-Sab 

-4cX     3a=—12ac 

'-4:cX—4:b=+16bc, 

We  attach  a  meaning  to  these  operations  on  single  terms, 
after  the  same  manner  as  in  Addition  and  Subtraction. 
Thus,  the  statement  — 4cX~45=  +  16^c,  means  that  if  —ic 
occur  among  the  terms  of  a  multiplicand,  and  —4^  among 
the  terms  of  a  multiplier,  there  will  be  a  term  i-lQic  in  the 
product  corresponding  to  them. 

As  particular  cases  of  examples  of  this  sort,  we  haye, 

2aX-4=-^8a,  2X-4=~8,   -2X-l=+2. 

Eemark. — If  several  single  terms  are  to  be  multiplied  together  the 
product  will  be  -f  or  — ,  according  as  the  number  of  negative  factors  is 
even  or  odd. 

Thus,  4aX-25x     Zc  is  -^idbe 

AaX-2hX—oc  is  -\-24:abo 
4aX-2bX-ScX-2d  is  -iSabcd. 

63.  The  rules  for  Multiplication  may  now  be  conveniently 
presented  thus : 

To  multiply  single  terms:  Multiply  together  the  numeri" 
cal  coefficie7its,  init  the  literal  factors  after  this  product,  and 
determine  the  sign  hy  the  Rule  of  Signs, 

To  multiply  quantities  of  two  or  more  terms:  Multijjly 
each  term  of  the  multiplicand  hy  each  term  of  the  multiplier, 
according  to  the  ride  for  single  terms,  and  the  sum  of  tJicse 
separate  products  will  be  the  product  required. 

The  process  is  generally  conducted  as  in  the  following 
examples. 

Two  General  Rules  of  Multiplication. 


EXAMPLES.  48 


Ex.  1.    Multiply  -2a'b''+     5ab^     -     7b* 

by  — 4«& 
Prod.         8a'b''-20a''b^+2Sab\ 

Ex.  2.    Multiply  2a+db-ie 

by  a+  5—  c 


Prod,  by      a  =  2a''+Sab-4:ac 

by  +5=        +2ab  +  Sb''^Abc 

by  -c  =  —2ac-dbc-\-4:G'^ 


Whole  prod.  =  2a^  +  5a6  -  6ac  +  db''  -  7bc  +  4c^ 

rix.  3. 

a  +  b               Ex.  4.    «  +  &               Ex.  5.    cJ-5 

a+b                             a—b                             a-b 

a'  +  ab                         a'  +  ab                           a'-ab 

+  ab  +  b''                      -ab-b^                        -ab  +  P 

a''  +  2ab  +  b\                «'    ^^    -b\                    a^~2ab  +  b\ 

Ex.  6. 

x  +  a               Ex.  7.    x^  +  {a  +  b)x  +  ab 

x  +  b                              x  +  c                                           y^ 

x^  +  ax                           x^  +  {a  +  b)x^  +  abx 

+  bx  +  ab                        +         cx''  +  (ac  +  bc)x  +  abc 

Ans, 

x^  +  {a  +  b)x  +  ab           x^  +  {a+b-\-  c)  x^  -^{cib^roc-^  be)  x  +  abc 

Ex.  8, 

x^—ax'^+     bx     —     c 

x^  +  mx  +      n 

x^—ax'^+     bx^    —   cx^ 
+  mx'^  —  amx^   +  bwtx^ — cmx 

+  ?zrt^  — aiix^     +bnx—cn 


Ans.      ic'^  —  (<x — m)  x"^  +  {b— am  +  ?i)«""*  —  (c — 6//i  +  a»)  a?^  —  {cm—bn)x — en . 

64.  We  arrange  the  terms  of  the  partial  products  so  that 
like  terms  may  stand  in  the  same  column.  This  enables  us 
to  collect  the  terms  easily,  in  order  to  get  the  final  result. 
With  the  view  of  bringing  the  like  terms  of  the  product  into 

The  order  of  Arrant^jcment  of  Terms  of  Multiplicand  and  Multiplier. 


44  ELEMENTAEY    ALGEBKA. 

the  same  column,  we  arrange  the  terms  of  the  multiplicand 
and  multiplier  in  a  certain  order.  "We  fix  on  some  letter 
which  occurs  in  many  of  the  terms,  and  arrange  the  terms 
according  to  the  poivers  of  that  letter. 

Thus,  taking  the  last  example,  we  fix  on  the  letter  x ;  we 
put  first  in  the  multiplicand  the  term  containing  the  third 
power  of  X ;  next  we  put  the  term  which  contains  the  next 
power  of  X ;  next  the  term  which  contains  the  first  power  of 
x\  and  last  we  put  the  term  which  does  not  contain  x  at  all. 
The  multiplicand  is  said  now  to  be  arranged  according  to  de- 
scending poicers  of  x.  We  arrange  the  multiplier  always  in 
the  same  way  as  the  multiplicand.  It  would  haye  done  as 
well  to  arrange  them  both  according  to  asce7iding  poivers 
of  X. 

Examples — 9. 

1.  Multiply  ax?\f  by  Ixy  ;  mx^  by  —nx^ ;  —acx  by  —2axi/; 

ale  by  Ic,  —ale  by  —ac]  x^y  by  —xy"^, 

2.  Multiply  x^—xy-^-y"^  by  x,  and  a^  —  ax-{-x^  by  —ax] 

x^ — ax-\-l)  by  —  alx ;  x^ — ox^y  +  ^xy""  —  %f  by  xy, . 

3.  Multiply  2«  +  Z'  by  a-^-U,  and  "la-l  by  c—M, 

4.  Multiply  "^x-V^  by  2:^+3?/,  and  3^5+4^'  by  2al-U\ 

5.  Multiply  x'  +  ^x—^  by  ^+3,  and  x''—4:X-\-^  by  x—2, 

6.  Multiply  a'^-^a-l  by  a'-a-\-l,  and  by  a'-3a-L 

7.  Multiply  2W-\-^x\j-\-^xy''  +  y'hjdx-y, 

8.  Multiply  a'-2a'l-^^a''l''-%al)'  +  Ul'hy  a-^^l. 

9.  Multiply  x"" -\-%ax-{-'^a^  hj  x'^—2ax-\-a'. 

10.  Multiply  9a'-3«J  +  Z?'-6a-2^+4by  3r«+J  +  2. 


GE]!s^EHAL    EESULTS    IX    MULTIPLICATION.  45 

1 1 .  Multiply  x'^-\-y'^+z'^  +  xy — xz + yz  by  x—y-]-z. 

12.  Multiply  a'-\-2a''-h2a  +  lhy  a'-2a''-{-2a-l. 

13.  Multiply  a'  +  W-i-9o''+2aI)  +  dac-6I)chj  a—2b-3c, 

14.  Multiply  a'-2a'b  +  3a'b'-2aF  +  I)'hja'  +  2ad  +  I?\ 

15.  Multiply  x^  —  ax  +  b  hj  x—c,  and  by  x^  +  ax—c, 

16.  Multiply  l  —  ax={-I?x^  —  cx''hjl+x—x''. 

IX.    General  Eesults  in  Multiplication. 

65.  N.B. — The  rules  for  the  management  of  BracTcets, 
given  in  YII.,  apply  only  to  the  addition  and  siiMractmi  of 
quantities  so  enclosed.  If  a  collection  of  quantities  within 
brackets  is  to  be  ^nultijolied  or  divided  by  any  quantity  or 
collection  of  quantities,  the  brackets  must  not  be  struck  out 
until  the  multiplication  or  division  is  actually  performed. 
Thus  {a-\-'b)x{c-\'d)  signifies  that  «  +  Z>  is  to  be  taken  c-\-d 
times,  and  is  obviously  not  the  same  as  either  a-\-'b{c-\-d),  or 
{a-\-'b)c-{-  d.  Again,  [a+'b)-^{c-{-  d)  is  not  equivalent  to  either 

a^-'b^{c-{-d),  OY\a-{-'b)-~c-{-d]  but  it  may  be  written -^, 

the  line  which  separates  the  numerator  and  denominator 
serving  as  a  vinculum  to  hoth. 

The  learner  would  do  well  to  practise  multiplication  of 
quantities  by  means  of  braclcets  as  early  as  possible. 

Thus, 

Ex.  1.     {a—l){c—d)^{a—'b)c—{a—'b)dy 

=ac—'bc—{ad—'bd), 

-  ac —be— ad  +  bd. 

The  Management  of  Brackets  In  Multiplication. 


4G  ELEMENTARY    ALGEBRA. 

Ex.  2.     (x-\-a){x+b)=^{x  +  a)x+{x-\-a)I?, 
= x^  +  ax + Ix + ad, 
^x'  +  {a  +  b)x  +  ah* 


Ex.  3.  {x+l){x-\-2){x+3)  =  {x'  +  2  +  l.x  +  2){x+3), 

=  {x'  +  dx+2)x+{x'  +  dx-\-2)3, 

=x'  +  3x'  +  2x  +  3x^  +  9x  +  6, 

=x'  +  6x''  +  llx  +  Q, 
Ex.  4. 
{a-]-b—c){a  +  b—c)  =  {a  +  h—c)a  +  {a-^h—c)h—{a  +  I?—c)c, 

=a'^  +  ah—ac+al)  +  d^—l?c--ac—dc  +  c^, 

=:o^-\-2al)-\-V-2ac-21)c-\-c\ 

66.  The  student  should  notice  some  results  in  Multiplica- 
tion, so  as  to  be  able  to  apply  them  when  similar  cases  occur, 
and  write  down  at  once  the  corresponding  products. 

Ex.  3,  Art.  63/giyes  {a-\-'b){a-\-'b)  or  {a^-iy 
=:a'+2a$4-Z>Xi).    Thus, 

The  square  of  the  siwi  of  two  qicantities  is  equal  to  the 
sum  of  the  squares  of  the  two  quantities  ijicreased  ly  tiuice 
their  product, 

Ex.  5,  Art.  63,  gives  {a—l){a—l),  or  {a—iy 
=.a^-2db^V  (ii).    Thus, 

Hie  square  of  the  difference  of  two  quantities  is  equal  to 
the  sum  of  the  squares  of  the  two  quantities  diminished  dy 
twice  their  product, 

Ex.  4,  Art.  63,  gives  {a  +  d){a-i)=a'-'b'  (iii).    Thus, 

The  product  of  the  sum.  and  difference  of  two  quantities  is 
equal  to  the  difference  of  their  squares. 

The  sqnare  of  the  sum  of  two  quantities.     The  pquare  of  the  difference  of  two 
quantities.    The  product  of  the  difference  of  two  quantities. 


MULTIPLICATION.  47 

67.  General  results  expressed  by  symbols,  as  in  the  equa- 
tions (i),  (ii),  and  (iii),  are  called  formulas. 

In  these  formulas,  a  and  h  indicate  any  quantities  or  ex- 
pressions whateyer. 
Bemark. — We  may  express  the  two  formulas, 

{a  +  hf=a''  +  Uh  +  ¥  ;  and  {a-hf-=a'' -2al)  +  b\ 
in  one  formula.    Thus, 

{a±bf=a''±2ab-{-b\ 

where  ±  indicates  that  we  may  take  either  the  sign  +  or  the  sign  — 
keeping  throughout  the  vpper  sign  or  the  lotcer  sign.  a±b  is  read  thu» 
"  a  plus  or  minus  Z>."     ±  is  called  the  double  sign. 

As  applications  of  these  rules  or  formulas,  we  have, 

(x  +  yy=x'-]-2xij-\-y%  {x-2y=x'-4.x  +  4:, 

{2x  +  yy=4:x''  +  4.xy+y\ 

{2ax-3hjy=4:aV-12aI?xy+9by, 

{29y={30-iy  =  900-60  +  l, 

(54)^^=  (50  +  4:)^=:2500 +400 -f  16=2916, 

{x-]-2){x-2)=x'-4:, 

{2ax-3hy)  {2ax-\-Uy)  =4aV-9Z>y, 

(127)'-(123)^=:(127  +  123)(127-123)=250X4 

==1000. 

68.  Ex.  6,  Art.  63,  gives 

(x  +  a) {x -{■l)=:x^-\-{a-\- h)x  +  ab  (iv), 

where  the  coefficient  of  x  is  the  sum  of  the  two  latter  terms 
of  the  factors,  x-ha,  x-\-b,  and  the  last  term,  -i-ah^  is  their 
product.    In  like  manner,  we  shall  have, 

(x^a)  (x—h)  =x^ — (a  +  d)x+ abf 

(x—a)  {x  +  h)  —x^  -f  {l)—a)x—ab, 

The  use  of  Fo!*mulas.    The  double  sign. 


48  ELEMENTARY    ALGEBRA. 

Thus, 

{x-5){x  +  2)=x'-{-{2-d)x-10=x'-  3.T-10, 
{x-Q){x-6)=x'-{6  +  6)x-hdO=:x'--llx  +  30, 
{xi-2){x-2){x  +  d){x-d)=z{x'-4:){x'-9), 
=x'-{9  +  ^)x''  +  36=x'-ldx'-\-3(j, 

69.  By  a  little  ingenuity  the  formulas  (i),  (ii),  (iii),  and 
(iv),  may  be  extensively  applied  to  lighten  the  labor  of  Mul- 
tiplication. 

Suppose  we  require  the  square  of  x  +  y  +  z.  Denote  x  -{-y 
by  a. 

Then  x  +  y-\-z=a-\-z]  and  by  the  use  of  (1)  we  haye, 

(a^-  zy  =  0"  +  2az+ z^=:{x  +  ijy  +  2{x-\-^j)z^-  z" 

^x'^  2xy  +rf  +  2xz  +  2yz  +  2;^ 

Thus,  {x-^ y  +  zy ^x" -^ if  +  z"  +  2xy  +  2yz  +  2xz, 
Suppose  we  require  the  square  oi p  —  q  +  r—s.   Denote  p—q 
by  a,  and  r—s  by  l]  then^  — ^  +  r— 5  =  a  +  Z>. 

By  the  use  of  (i)  we  have, 

{a^iy:=:a'  +  2al)  +  ¥={p-qy  +  2{p-q){r-s)-\-{:r-sy. 

Then  by  the  use  of  (ii)  we  express  {p—qy  and  {r—sy. 

Thus,  {p-q  +  r-sy 

=p''  -'2p)q^-  q"  -\-2{pr—ps—qr  +  qs)  -\-r'^  —  2rs-\-  s" 

—p"  +  g^  +  r""  +  /  +  2pr  +  2qs —2pq—2ps—2qr— 2rs. 

Suppose  we  require  the  product  of  p—q-\-r—s  and 
p—q—r  +  s, 

ljeip—q=a,  and  r—s^h;  then 

p—q-^-r—s  —  a  +  h,  and  p  —  q—r  +  s=a—b, 

Wliat  are  the  Formulas  (i),  (ii),  (iii),  and  (iv)? 


multiplicatio:n'.  41) 

Then  by  the  use  of  (iii)  we  have, 

and  by  the  use  of  (ii)  we  have, 

{p—q-\-r—s)  {2y—q  —  r-\-s)=p''  —  ^pq  +  q^—{r^—2rs-[-s^) 

=2o''  +  q'-r'-s'-22Jq  +  2rs, 

As  the  student  becomes  more  familiar  with  the  subject, 

he  may  dispense  with  some  of  the  work.     Thus,  in  the  last 

example,  he  will  be  able  to  omit  that  part  relating  to  a  and 

Z>,  and  simply  put  down  the  following  process : 

(p  —  q  +  r—s)  {p  —  q—r  +  s={p  —  q  +  {r—s))  {p—q—{r—s)), 
=:{p~qy-{r-sy 
—p"" — 2pq  -\-q^—  (r- — 2rs  +  s-) 
—p^ — 2pq  +  (^^ — r-  +  2r5 — 5I 


70.  Ex.  1.     {ax  +  l) -^ cy  +  dy={ax-\-h -\- cy  +  ciy, 

=  {ax-^I)y+{cy  +  dy  +  2{cfx-{-b){cy  +  d), 
= a^x^  +  ^'  +  2al?x  -f-  (fy^  +  <^  +  2cdy 

+  2acxy  +  2adx  +  2I)cy  +  2M. 
Ex.  2.     (a^—ax-hx^)  {a'^—ax  —  x'^)=^hj  (ii)  {a^—cfxy-^x* 

=  a^— 2a^x  +  a-x^ — x*. 
Ex.  3.     {a^-i-ax—x-){a'^—ax—x'^) 

=  { («^ — x")  -f  ax}  { (a^ — x"^)  —  ax} 
=z{a'^— ^-)  -  —  a^x^ =a^— 2a^x"  -\-x^~  a'x^ 
=  a'*  —  3a'x'^  +  x\ 

N.  B. — The  formula  here  em])]oyed,  {a-{-b)X{a—b)z=a'^  —  b'^,  may  be 
alwaj^s  applied,  whenever  it  is  seen  that  the  two  quantities  to  be  mul- 
tiplied consist  of  tirms  which  differ  only  (some  of  them)  in  sign,  by 
taking  for  a  those  terms  which  are  found  with  their  signs  unaltered  in 
each  of  the  given  quantities,  and  the  others  for  b.  Thus,  in  Ex.  3,  a? 
-■x^  appear  in  both  the  given  quantities,  whereas  in  the  one  we  havo 
-i-ax,  in  the  other  —ax\  hence,  the  product  required  is  {a^—x^f — a\-;^, 
as  a  Dove. 

3 


V  ele:.iextary   algebra. 

Ex.  4.     (a'  +  ax  +  x')  {a'-ax+x')  =  {cr+xy-a'x' 

=  a^  +  a^x^  -f  x^, 
Ex.  5.     {a'^+ax—x-){a-—ax  +  x^)  =  a*~{ax—xy 

=^a*--a'x'^+2ax^—x\ 
Ex.  6.     {ar—ax-\-x^^)  {ax  +  x^—a")=x^  —  {a^—axy 

=x^—a^-\-2a^x—a^af, 
Ex.  7.     (a  +  b  +  c  +  d)  {a  +  b-c-cl)  =  {a  +  by-{c-{-dy 

=za'  +  2al)  +  ¥-  &  -  2cd-  d\ 
Ex.8.     {a^2h-?>c-d){ci-%b-^U-d) 

=  {a-dy-{:2b-^cy 

=a''-2ad+d'-4:b'  +  12bc-9c\ 

Examples — 10. 

1.  Write   down   the    squares   of    a—x,   l  +  2ic'',   2a* +  3 

3x—4:i/, 

2.  Write  down   the   squares  of  3  +  2x,  2x—3y,  a^—3axw 

bx^  —  cxy, 

3.  Write   down   th©  products  of  {2a-\-l)x{2a—l), 

{3ax-{-b)x{3ax-b),  {x-~l)  {x^l)  {x'-^l). 

4.  Write  down  the  products  of  (a:  +  3)  X  (^  +  1), 

{x''-\-4)x(x'-l),  {ab-3)  {ab-\-2y 
(2ax—3b)  {2ax-b), 
6.    Find  the  continued  product  ofx-{-a,  x—a,  x+2a, 
and  x—2a. 


EXAMPLES.  51 

n.     Obtain  the  product  of  mx  +  2ny,  mx~27iy,  mx—Sny, 
and  mx-\-3ny, 

7.  Simplify  3{a-2Q:y  +  2{a-2x)  {a  +  2x) 

+  {3x—a)  {3x  +  a)  —  {2a-3xy, 

8.  Multiply  x''  +  2xy  +  2^'  by  x^ — 2xy  +  2y\ 

and  2a^  -  3ah  +  6'  by  2a'  4-  3ab  +  Z>'. 

9.  Multiply  a-\-h-{-c  by  a-^h  —  c^  by  a— 5  +  c, 

and  by  a—h—c. 

10.  Multiply  a— ^  +  c  by  a— 5--C,  by  5  +  c— flj, 

and  by  c—h—a. 

11.  Multiply  2a+5— 3c  by  2a— ^  +  3c,  and  by  h  +  'dc~2a. 

12.  Multiply  2a  —  h—3c  by  2a  +  ^>  +  3c,  and  by  Z*  — 3c— 2a. 

13.  Multiply  a  +  5  +  c+c?  by  a—h-^c—d,  by  a—h—c-\-d, 

and  by  b-^c—d—a, 

14.  Multiply  a— 2^  +  3c  +  6^by  a  +  2^-3c+<^, 

by  25  — a  +  3.c  +  <^,  and  by  a  +  25  +  3c— ^. 

71.  There  are  other  results  in  Multiplication  which  are  of 
less  importance  than  the  four  formulas  given  in  Art.  QQ,  but 
which  are  deserving  of  attention.  We  place  them  here  in 
order  that  the  student  may  be  able  to  refer  to  them  when 
they  are  wanted ;  they  can  be  easily  verified  by  actual  multi- 
plication. 

{a'-b){a'-\-ab-hb')=a'-b% 
{a+by=z{a  +  b)  (d'+2ab  +  ¥)  =  a''\-3a'b-i-3ab'  +  b% 
(a-by={a-b){a'-'2ab-{-b')^a'-3a'b-{-3ab'-b\ 

other  results  in  Multiplication. 


52 


ELEMEXTAIIY    ALGEBKA. 


X.       DlYi:,ION. 


72.  Division,  as  in  Arithmetic,  is  the  inverse  of  Multipli- 
cation. In  Division  we  have  given  the  product  and  one  of 
the  factors,  and  we  have  to  determine  the  other  factor.  The 
factor  to  be  determined  is  the  quotient, 

73.  I.  The  rule  for  the  division  of  simple  expressions 
follows  at  once  from  the  corresponding  case  in  Multiplica- 
tion. 

For  example,  we  have. 


therefore        — - 


therefore 


therefore 


therefore 


l^ahc 


4.^^"'^ 


12ahc 


■Aab. 


Also       ^abX  —3c  =  —  12al)c; 
-12al?c 


12abc 

—  =  — dc, 


4,ah 
Also  —  4:011)  X 
-VZahc  _ 

Also  --4a^X— 3cj= 


^iat. 


=^-4.ab. 


I2abc_ 
-iab 


=  -3c, 


12ahc_ 
-3c 


=  -^ab. 


Hence,  we  have  the  following  rule  for  dividing  one  singte 
term  by  another  :  Divide  resjjedively  the  coefficient  and  literal 
parts  of  the  dividend  ly  those  of  the  divisor ;  and  then,  if  the 
tivo  quantities  have  like  signs,  prefix  to  the  quotient  the  sign 
+,  if  different,  the  sign  — . 


What  is  Division?    Rale  for  dividing  one  single  tenn  by  another. 


DIVISION.  53 

This  division  is  the  familiar  process  of  cancelling  like  fac- 
tors in  Arithmetic.  Hence  the  rule  may  be  given  briefly 
thus : 

Strike  out  from  the  dividend  the  factors  which  occur  in  the 
divisor  ;  the  rest  of  the  dividend  is  the  quotient,  ivhose  sign  is 
deternmied  hy  the  Rule  of  Signs,  Viz,  \  Like  sigjis  give  +. 
unlike  sig7is  — . 

Thus, 

-7^-v-5=-7,  -ax-^a^x,  Uab-^n=2a,  7b^n=l, 

abc~ab=c. 

74.  One  poiuer  of  a  quantity  is  divided  by  another  i)oioer 
of  the  same  quantity  by  subtracting  the  index  of  the  latter 
from  that  of  the  former. 

For  example,  suppose  we  have  to  divide  a^  by  a^ 


a'    a\a'      . 

'=a'-\ 

Or  we  may 

■  show  the  truth  of  the  rule  thus : 

a'Xa'. 

=a\ 

Therefore, 

a'      3 

And  generally,  if 

7n  and  n  be 

positive 

integers. 

and  m  >  n 

fl^^'-f-a"- 

^a'''-^ 

Similarly, 

Za'l)' 

=  2a'b% 

xhfz' 

z^x'yzK 

xyz 

ha'b\ 
d'bc 

(=5a'b'c% 

a^'b' 

_^m-pyn-q^ 

Division  of  one  power  of  a  quantity  by  another  power  of  the  same  quantity. 


54  ELEMENTARY     ALGEBRA. 

75.  It  may  happen  that  the  factors  of  the  divisor  do  not 
occur  in  the  dividend.  In  this  case  we  can  only  indicate 
the  division.  Thus,  if  6a  is  to  be  divided  by  3c,  the  quo- 
tient is  indicated  by  6a-^3c,  or  by  — . 

oC 

Again,  it  may  happen  that  some  of  the  factors  of  the  di- 
visor occur  in  the  dividend,  but  not  all  of  them ;  or  that  a 
power  of  a  quantity  occurs  in  the  dividend,  and  a  higher 
power  of  the  same  quantity  in  the  divisor.  In  this  case  the 
indicated  quotient  is  a  fraction,  which  can  be  simplified  by 
striking  out  common  factors,  as  in  Arithmetic. 

Suppose,  for  example,  16a^b  is  to  be  divided  hjQbc;  we  have, 

loa'b_  5a'xSb  _5a^ 
6bc  ~  2cX3b    ~  2g' 

striking  out  the  common  factor  3b. 

Again,  if  4tab'^  is  to  be  divided  by  3cb%  the  quotient  is  indi- 

cated  by  ^r-n.-     Eemove  the  factor  b"^  which  occurs  in  both 

dividend  and  divisor, 

^aW  _  ia 
'3cb'~'3cb^' 

76.  II.  To  divide  a  quantity  consisting  of  two  or  moro 
terms  by  a  single  term : 

Divide  each  term  of  the  dividend  by  the  divisor,  and  collect 
the  results  to  form  the  complete  quotient. 
For,  since  fl^+^— c  +  &c.  multiplied  by  m  produces 
ma-hmb—mc-{-&c., 
/.  ma-\-mb—7nc+&o.  divided  by  m,  gives 

a  +  b—c-i  &c. 
Hence,  the  rule  is  as  above  stated. 

Rale  for  Case  n. 


DIYISIOX.  55 

Ex.  1. — h— =4:a'-3Z>c4-a. 

a  a         a       a 

a^x'^  —  6aJ)x^  +  6ax^_a^x'^     6abx^     (5ax* 
aa;  ax       ax        ax 

=  a'-6I)x  +  6x\ 

Ex.  3.     (a+I)+c)-~abc=:-T-+-Y~-\--j-^Y--h — h- 1^. 
^  ^  aoc     ado     abc     be     ac    ab 

^      ,      a'c'-2abc'  +  dac'  a'c'      2abc'     3ac' 

Ex.  4     T-r-a =  — 7-T-2  +  - 


-4:abc^  ^abc^     4:abc^    4iabc^* 

__^     l_3c 

77.  III.  To  divide  one  expression  by  another  when  tne 
divisor  consists  of  two  or  more  terms^  we  must  proceed  as  in 
the  operation  called  Long  Division  in  Arithmetic.  The  fol- 
lowing rule  may  be  given : 

Arra7ige  both  dividend  and  divisor  aecording  to  ascending 
poiuers  of  some  common  letter,  or  both  according  to  descending 
poiuers  of  some  common  letter.  Divide  the  first  term  of  the 
dividend  by  the  first  term  of  the  divisor,  and  ][tut  the  result 
for  the  first  term  of  the  quotient ;  multiply  the  luhole  divisor 
by  this  term  and  subtract  the  product  from  the  dividend.  To 
the  remainder  join  as  many  terms  of  the  dividend,  tahen  in 
order,  as  may  be  required,  and  repeat  the  whole  operation. 
Continue  the  process  until  all  the  terms  of  the  dividend  have 
been  tahen  doivn. 

The  reason  for  this  rule  is  the  same  as  that  for  the  rule  of 
Long  Division  in  Arithmetic,  namely,  that  we  may  break 
the  dividend  up  into  parts  and  find  how  often  the  divisor  is 
contained  in  each  part,  and  then  the  aggregate  of  these  re- 
sults is  the  complete  quotient. 

Rule  for  Case  III. 


56  ELEMENTARY    ALGEBRA. 

78.  We  shall  now  give  some  examples  of  Division  arranged 
in  a  convenient  form. 

Ex.  L     1— .t)  l-2x-i-x'  {l~x 
1—  X 

—  x  +  x^ 

—  x  +  x^ 


Ex.  2.     3x  ^ %)  Gx'  - 1 7x'ij  + 1  (jy\2x'  -  dxi/ - 4.f 
-  9x'y-j-l2x?/ 


-12xf-{-lGy' 

Ex.  3.     a—x)  a^—x''  {a^^ax^x^ 
a^—ct^x 


d\v- 

-x' 

cc'x- 

-  ax^ 

a/jy  — 

■x^ 

ax'-^ 

x' 

Ex.  4.     a-^x)  d-\-'^  {a^—ax  +  x'^ 
a^  +  a'^x 
—cC'x-^x^ 
^a/x—ax^ 

ax^  +  x^ 
aaf  +  x^ 

Ex.  5. 

-2^^4-3^=)  na*-10a'b-22aW  +  22aF-{-lDl?'  {da' -4.ab  +  5d' 
3a'-  6a'b+  9a'b' 

-  4.a'b  +  rda'b'-22ab' 

-  4a'b+   8aW-12ab' 

baW-lOab'  +  lob' 
da'b'-lOab'i-lDb' 


Consider  the  last  example.  The  dividend  and  divisor  are  both  ar- 
ranged according  to  descending  powers  of  a.  Tiie  first  term  in  the 
dividend  is  3a*,  and  the  first  term  in  the  divisor  is  a"^ ;  dividing  the 
former  by  the  latter  we  obtain  Sa^  for  the  first  term  of  tlie  quotient. 
We  then  multiply  the  whole  divisor  by  3a- ,  and  place  the  result  so 


DIVISION.  57 

lliat  each  term  comes  below  the  term  of  the  dividend  which  contains 
the  same  power  of  a;  we  subtract,  and  obtain  — 4a^b  +  lda^b'^ ;  and  wo 
bring  down  the  next  term  of  the  dividend,  namely,  — 22a&^.  We  di' 
vide  the  first  term,  —Aa'^b^  by  the  first  term  of  the  clivisor,  a'^ ;  thus  we 
obtain  —  4:ab  for  the  next  term  in  the  quotient.  We  then  multiply  the 
whole  divisor  by  —  4«Z>  and  place  the  result  in  order  under  those  terms 
of  the  dividend  with  which  we  are  now  occupied ;  we  subtract,  and 
obtain  5a'^b'^—10ab^ ;  and  we  bring  down  the  next  term  of  the  dividend, 
namely,  15^^.  We  divide  6a'^b'^  by  a\  and  thus  we  obtain  5^^  for  the 
next  term  in  the  quotient.  We  then  multiply  the  whole  divisor  by  55^, 
and  place  the  terms  as  before ;  we  subtract,  and  there  is  no  remainder. 
As  all  the  terms  in  the  dividend  have  been  brought  down,  the  opera- 
tion is  completed;  and  the  quotient  is  Sa"^ — 4:ab-hob'\ 

It  is  of  great  iinj^ortance  to  arrange  ioth  dividend  and  di- 
visor according  to  the  same  order  of  some  conamon  letter  ;  and 
to  attend  to  this  order  in  every  part  of  the  operation, 

79.  It  may  happen  that  the  division  cannot  he  exactly 
performed.  Thus,  for  example,  if  we  divide  <^^+2«^-f2Z>^  by 
a-^1)  we  shall  obtain  a+J  in  the  quotient,  and  there  will 
then  le  a  remainder,  V,  This  result  we  place,  as  in  Arith- 
metic, in  the  quotient  over  the  divisor,  in  the  form  of  a 
fraction,  thus  indicating  that  If  remains  still  to  be  divided 
by  a+h.    Thus, 

-T — =a  +  Z>H — -y. 

Ex.  6.    a+x)  c^^rx^  {a—x^ 


a+x 
a^  +'  ax 


—ax  +  x 

—ax—x^ 


2x' 

2x^ 


Ex.  7.    a-x)  a^^x^  {a^x-\- 


a—x 
-ax 

ax+  x^ 
ax—  x^ 

2? 


Important  principle  in  Division.    Division  with  a   remainder. 


58  ELEMENTARY    ALGEBRA. 

80.  We  give  some  more  examples : 
Ex,  8.     1-^)1       {li-x+x^  +  x^  +  &c,+  -^~^ 


-X 


+  x 
+  x—x^ 


+  X'' 

+  x''-x' 

+  x'-x* 
+  x'  &c. 

Ex.  9.   Divide  x'-5x'  +  '7x'-{-2x'-6x-2  by  l  +  2x—3x'  +  x\ 
Arrange  both  dividend  and  divisor  according  to  descend- 
ing powers  of  x. 
x'-3x''-\-2x-]-l)  x'-bx'  +7x'-^2x'-6x-2  (x''^2x--2 


x''-3x'  +  2x'+  x' 

--2x'-2x'  +  6x'  +  2x'-Gx 
—2x'          +6x'-4:x'-2x 

-2x'          +6x'-4.x- 
-2x'          +6x''-4:X- 

-2 
-2 

Ex.  10.   Divide  64-^"  by  2- 


-a. 


64-32a 


33a- 

-a' 

32a- 

-16a' 
16a'- 
16a'- 

-a' 
-8a' 

8a'- 
8a'- 

-a' 

-ia' 
4a*- 
4a'- 

-a' 
-2a' 
2a'- 
2a'- 

-a' 
-a' 

DIYISIOIS^  59 

Ex.  11.     Divide  a^  ^h^-^-c^—^^aU  by  a  +  ^  +  c. 

Arrange  the  diyidend  according  to  descending  powers  of  «, 

a  +  'b  +  c)a'                  -?>abc-\-¥  ^c\a'  -ab-ac-^F  -U-^G^ 
a^-^a'h  +  a'c 

—a^h—a^G  —?>abG 


—c^G    ■\-a¥  —  1abG 

—a^G  —  obG—aG^ 

aW—  abc  +  aG^  +  h^ 
a^ +h'  +  l)'G 

—  abc^a&       —Fg 

—  abc  —Wg—W 

a?  '\-M-\-g'' 

ac^  +Ig''-\-g' 

The  above  is  the  easier  method  in  such  a  case,  but  the  fol- 
lowing, in  which  the  coefficients  of  the  different  powers  of  a 
are  collected  in  brackets,  is  the  neater  and  more  compendi- 
ous: 

—  {b  +  G)a'-dI)Ga 

-{b  +  G)a'-{b'+2bc+G')a 


+  {b'-  bG+G')a-^{b'  +  G') 

+  {b'-    bG+G')a+{b'  +  G') 


Examples — 11. 
Divide 

1.  16x'  by  ^x\         2.  24a°  by  -Sa\        3.  182:y  by  6<y. 
4.  2^.a'b'G'  by  -da'b'G\  5.  20a'b'xY  by  5Z»Vy. 


^^  ELE31EOTAKY    ALGEBRA. 

6.  4^^-  8:^^  +  16:.  by  ^x,  7.  3^^-12^^  +  15a^  by  ~-U\ 

8.  xhj-3xY  +  4:xy'hjxy, 

9.  -15^^Z*^-3a^Z^^+12a^  by  ^3a^. 

10.  60r.^^V-48a^^V  +  36«T.^~20«5^^  by  ^ahc\ 

11.  ^^^-7^  +  12  by  :.-3,  12.  ^^  +  r.^72  by  .'.  +  9. 

13.  2x'-x'  +  ^x-^hj2x~d. 

14.  Qx'  ^Ux'^  4.x  +  24  by  2x  +  6. 

15.  9x'  +  3x'  +  x-~lhj3x~l. 

^^'  7x'~24.x'-{-58x~21hj7x~3. 

17.  ^«-l  by  x~l.  18.     a«-2a^^+^^  by  a~^. 

19.  ^'^-81^^by:r-3y 

20.  x'~-2x\j-\-2x\f-xf  by  x~y. 

21.  ^^_y^  by  a;-y.  22.  a^+32^^  by  a +  25. 

23.  2a'-V%7aF-%Wh^a^3-b. 

24.  ^^  +  :^V  +  ^y +  2;y +  :^3/^  +  ,y^  by  r^^/. 

^h.  x'  +  2x^y+3xY^xY-2xf-~3f  by  o.^-.^^ 

26.  ^'-5a5^  +  lla3^-12a;  +  6byi«^-3^  +  3. 

27.  x'  +  x'^^x''-^Ux-4.hjx'!-\-4.x  +  L 

28.  aj*-13a3H36  hj  x'-^^x-\-Q. 

29.  i«*  +  64by^H4.T  +  8. 

30.  x'  +  10x'  +  3bx'  +  b0x  +  2i  by  ^^^  +  5^  +  4. 

31.  ^'-(a  +  5  +  £?)a3^  +  (a5-l-^^  +  ^^)^_^^^^ 

32.  ^'~{a+p)x''+{qj^ap)x-aqhj  x-a, 

33.  v' - ^?z?/^  +  ny"" — ny^ ^my-lhj  y-^1. 


FACTOKS.  61 

34.  a^—lf  +  c^  +  ^ahc  by  a—h-{-c,  and  a'—F—c^-dadc 

by  a—h—c. 

35.  a  by  1  + 33,  and  l-f2a;  by  1— 3;^',  each  to  four  terms  in 

the  quotient. 

36.  1  by  1— 2i;c+.T^,  to  four  terms. 

XI.    Factors. 

81.  We  shall  now  notice  some  general  results  in  Division, 
m  connection  with  those  already  given  in  Multiplication; 
and  we  shall  apply  some  of  these  to  find  what  expressions 
will  divide  a  given  expression,  or,  in  other  words,  to  resolve 
algebraic  exjjressions  into  their  factors, 

82.  For  example,  by  the  use  of  formula  (iii)  of  Art.  66, 
we  have, 

a'-Jj'-^^a'  +  h')  {a'-h'')^{a''  +  h'')  {a-^h)  (a-h) 
a'^I?'=--{a'  +  h')  {a'-b')  =  {a'  +  b')  {a'  +  b')  (a^b)  {a-b). 
Hence,  Ave  see  that  a^—b^  is  thQ  product  of  the  four  fac- 
tors a^-\-b\  a^-^y,  a  +  b,  and  a  —  b.     Thus,  a^  —  b^  is  divisible 
by  any  of  these  factors,  or  by  the  product  of  any  two  of 
them,  or  by  the  product  of  any  three  of  them. 
Again,  in  Art.  70,  we  have, 

{a'^ab  +  b'){a'-a])^b')  =  {a'^-by^{aby=a'-^a''b'-Vb\ 
Thus,   a^^a^b'^-^b"   is   the    product   of    the  two   factors 

€^-\-ab^-W    and   a''—ab-\rb'',  and   is    therefore    divisible    by 

either  of  them. 

83.  The  following  results  in  Division  may  be  easily  veri- 
fied, and  will  enable  us  to  write  out  with  ease  the  quotients 
in  many  similar  cases. 

x—y 


-y 


-1, 


How  to  resolve  Algebraic  quantities  into  their  Factors. 


62  ELEME^^TARY    ALGEBRA. 

X^  —  lt 

x-y  ^' 


Also, 


x-y 


x'-y' 
x-\-y  ^' 

—  =x  —  X  y  +  xy  —  y\ 

x  +  y  ^      iJ      J  ^ 

—  ^  —x''—x^y-^x^if—x^y^-\-xy''—y^j 


x-\-y 
and  so  on. 
Also, 

x-\-y      ' 

"^^^.x'-x'y  +  xY-xf^f, 
and  so  on. 

The  student  can  carry  on  these  operations  as  far  as  \w 
pleases,  and  he  will  thus  gain  confidence  in  the  truth  of  the 
statements  which  we  shall  now  make,  and  which  are  strictly 
demonstrated  in  larger  works  on  Algebra.  The  following 
are  the  statements : 

x'^^y^  is  divisible  by  x—y  if  n  be  any  whole  number; 
jc"— ?/"  is  divisible  by  cc  +  ^  if  w  be  any  even  whole  number; 
iK"  +  ^"  is  divisible  by  a; +^  if  ^  be  any  odd  whole  number. 


FACTORS.  G3 

"VVe  might  also  put  into  words  a  statement  of  the  forms  of 
the  quotient  in  the  three  cases;  but  the  student  will  most 
readily  learn  these  forms  by  looking  at  the  above  examples, 
and,  if  necessary,  carrying  the  operations  still  farther. 

We  may  add  that  x'^^'if  is  never  divisible  by  x-\-y  or  x—y^ 
when  n  is  an  even  whole  number. 

84.  The  student  will  be  assisted  in  remembering  the  re- 
sults of  the  preceding  Article  by  noticing  the  simplest  case 
in  each  of  the  four  results,  and  referring  other  cases  to  it. 
For  example,  suppose  we  wish  to  consider  whether  x^—y'^i'^ 
divisible  by  x—y  or  \i^  x^y\  the  index  7  is  an  06?^  whole 
number,  and  the  simplest  case  of  this  kind  is  a:— y,  which  is 
divisible  by  x—y,  but  not  by  x-{-y\  so  we  infer  that  x'—y'' 
is  divisible  by  x—y  and  not  by  x  +  y.  Again,  take  x*—y^\ 
the  index  8  is  an  even  whole  number,  and  the  simplest  case 
of  this  kind  is  x'—y'',  which  is  divisible  both  by  x—y  and 
x-{-y\  so  we  infer  that  x^—y^  is  divisible  both  by  x—y  and 
%-\-y, 

Now,  in  every  case  the  quotient  will  consist  (as  above. 
Art.  83)  of  terms  in  which  the  exponents  of  x  decrease,  and 
of  y  increase  continually  by  1 ;  but  when  the  divisor  is  x—y, 
these  terms  are  all  plus;  when  it  is  x-^-y,  they  are  alter- 
nately +  and  — . 

We  shall  now  apply  these  results  to  some  examples. 

Thus, 


'lax  —  l 


x  +  oy 
x'-lQ 


r=x'-3xy+9y\ 


x-2 


=  x'-2x^-h^x- 


Law  for  the  esponents  and  signs  of  the  quotient 


64  ELEMEXTAliy    ALGEBRA. 

Examples — 12. 
Divide 

1.  a^—x^  by  a-\-x,  a^—x^  by  a—x,  and  a^—x^  by  a-^x. 

2.  9^'-- 1  by  Zx-1,  252:'-l  by  hx-\-\,  and  4a;'-9 

by  2:^+3. 

3.  9wV  — 25  by  3??^?^+5,  and  16m*— 7^*  by  Am^  +  n^. 

4.  1  +  82;'  by  1  +  2^;,  27^'-l  by  3:r-l,  and  l-16a;* 

by  1+2:?:. 

5.  :r*-8l3/'  by  x-Sy,  a''+32h'  by  «+2Z>,  and  x''~-t/^ 

by  :^;'  +  ^^ 

6.  ia^  +  h^  by  i^+5,  and  x*y*—z'  by  0^^  +  ;^. 

7.  («  +  &)'  —  <?'  by  a  +  h—c,  and  a^—(l)'-cY  by  ^— Zi  +  (7. 

8.  (^+^)'+^^  by  .':c  +  y+^,  and  x^  —  {ij—zf  by  a;— ?/+^. 

85.  The  above  results  and  those  of  {^^)  may  also  be  ap- 
plied to  resolve  algebraical  quantities  into  their  elementary 
factors,  a  process  which  is  often  required. 

Ex.  1.  ^x^-if^{%x^y)  {^x-y), 
Ex.2.  .TH8  =  (a;  +  2)0'?;'-2:r  +  4). 
Ex.  3.  {2a-'by-{a-21)Y=^{U-'b^a-'21)){2a-l-a  +  2h) 

Ex.4.   x'-a'={x'  +  a'){x'-a') 

=z{x-{-a)  (x^  —  ax-ha^)  (x—a)  (x^  +  ax  +  a^), 
Ex.  5.   {a'-xy={{a-x)  {a'+ax  +  x')y 
=:{a-xyx{a'i-ax  +  xy. 


FACTORS.  Go 

Examples — 13. 
Eesolve  into  elementary  factors 

1.  l-4a;^  a'-dx',  9r}f-hi\  25aV-4<  16xy~2Dxy. 

2.  x'  +  y%  x'-y\  l  +  x'y\  x'-l,  a'xy'-x'y,  2a'b'c-SabV. 

3.  25x'-a'x%  a'-9a'b%  8^^-27,  a'-Sb%  aVy-{  27a;y. 

4.  x'  +  32,  aV  +  27x%  8x'-{-y%  a'h''-c\  a'bc  +  2a'I)c''  +  abG' 

5.  81:?;*-1,  x'-G4:,  x'-2baf  +  bV,  x'-2a'x'-\-a'x\ 

6.  (3.T-2)^~(2;-3)^  («  +  ^)'-4Z^',  (4a3  +  3^)^-(32:  +  4^)^ 

7.  {x'  +  yy-^xY,  c'-ia-by,  {2a  +  by-{2a-by. 

8.  a;'  +  '?/'  +  3a:?/  (x  +  y),  m'— if —in  {m^—n'')+ 72  {m—iiy, 

a'-ab  +  2  {b'-ab)+3  {a'-b')-4.  {a-by. 

9.  5  (x'-y')-\-'^  {x  +  y)\  3  {x'-y')-b  {x-yy, 

(^4.^)^  +  2  (:.^+:r^)-3(a:^-y^). 
10.  2{a'  +  a:'b-{-ab')-{a'-F),  a'-b'-3ab  {a-b), 
a'-b'  +  {a'-by-2a'  +  2a'b\ 

86.  So,  too,  Ave  may  often  apply  (68,  iv)  to  resolve  a  trino- 
mial into  factors  when  it  is  of  the  form  ax"^ -{- bx  +  c.  We 
repeat  formulas  (iv) : 

x''-\-{a  +  b)x-\-ab  =  {x  +  a){x-^b); 

x'^—{a-\-b)x  +  ab—{x—a){x—b); 

x'^  +  {a—b)x—ab={x  +  a){x—b). 

Ex.  1.  x''+7x+12  =  {  x  +  3){  X+4-). 

Ex.  2.  x'-dx  +  U^i  x-2){  x-^iy 

Ex.  3.  x'-DX-U^i  x-'7){  .T+2). 

Ex.4.  6.r'4.^-12  =  (3.7;-4)(2:^-f3). 

Resolution  of  Trinomials  into  factors. 


66  ELEMENTARY    ALGEBIIA. 

The  student  may  notice  that,  if  the  lasfc  term  of  the  given 
trinomial  be  positive  (Ex.  1,  2),  then  the  last  terms  of  the 
two  factors  will  have  the  same  sign  as  the  middle  term  of 
the  trinomial ;  but  if  negative  (Ex.  3,  4),  they  will  have,  one 
the  sign  +,  the  other  — . 

In  Ex.  4,  it  is  clear  that  the  first  terms  of  the  two  factors 
might  be  6x  and  x,  or  '^x  and  2.^•,  since  the  product  of  either 
of  these  pairs  is  6x^,  and  so  the  last  two  terms  might  be  12 
and  1,  6  and  2,  or  4  and  3 ;  it  is  easily  seen  on  trial  which 
are  to  be  taken,  that  is,  which  serve  also  to  produce  the 
middle  term  of  the  trinomial. 

Examples — 14. 

Eesolve  into  elementary  factors 

1.  a;'  +  6:?:  +  5,  2;'  +  9:r  +  20,  x'-hx-^Q,  x'-^x-Vl^, 

x''  +  ^x  +  'l,x'-l^x-\-^. 

2.  ir'+^-6,  ^'-a;-6,  a;'~22;-3,  x^-\-2x-lb,  x'^-lx-^, 

x'-^x-^. 
\  3.  4a;'  +  82;  +  3,  42;'  +  13a:  +  3,  42;'  +  lla;-3,  4:x''-4:X-3, 

dx'  +  4^x-4.,  6x''  +  6x-4c. 
4.  12^-'-5:z;-2,  12:i;'-14:z;  +  2,  12a;'-a;-l,  a;'-f  ^-12, 

dx^-2x-6, 
6.  aV-3a'x  +  2a\  a'-a^x-6ax\  3a*I)+a''b^-2ad\ 

12a'  +  aV-x\ 
6.  2x'y-^bxy-j-2xf,  9xYSxf-6y\  6aV  +  a'a;-a% 

(jb'x'-7bx'-dx\ 

87.  We  shall  now  give  a  few  examples  of  multiplication 
and  division  of  expressions  in  which  factors  or  terms  occur 
with  letters  for  their  exponents. 


GREATEST   COMMOX    DIVISOR.  07 

Ex.  1.  ]\Iultiply  a'-'^'F^+'o'  by  w"''-'¥c, 

Ans.    a'"'Z>'^+V. 
Ex.  2.  Multipljof^  +  fl^ii;'^ -a':^^  by  a"^^^^. 

Ans.    a"^a;'^  +  a"^+'a;'^--a"^+V» 
Ex.  3.  Multiply  a'-'l-d!'--lf^aV'-^  by  a  J. 

Ans.     oJ'W-aJ'-^l^^a^lf. 
Ex.  4.  Find  the  continued  product  of 

Ans.     a"+^Z»^^+V+^(^"+^. 
Ex.  5.  Multiply  a"*— 2c'*  by  a"*— c^ 

Ans.    a-'**— 3a"'c"4-2c-". 
Ex.  6.  Divide  115a'"^V^+'^J-'-'  by  -69a"^>V+'^d 

Ans.     -\dr-''c''dr'-\ 
Ex.  7.  Divide  a"'5-^"^-*^-  +  a"^--^^'-«'^-'^'  +  a"^-'Z>' by  fl^Z». 
Ans.     oJ^-^  -  a^'-l  +  oT-^V'  -  oT'-'W  +  ^'"-'Z^'. 
Ex.  8.  Divide  aj'^+^  +  a:"^^^.^"^^ +  /'*+«  by  a;"  +  ?/^ 

Ans.    a;"'^-?/". 
Ex.  9.  Divide  c^-"^-3a'"c^  +  2r«  by  fl^'^-c^ 

Ans.    a'"-2c". 

XII.    Greatest  Common  Divisoe, 

88.  In  Arithmetic,  a  whole  number  which  divides  another 
whole  number  exactly  is  said  to  be  a  divisor  or  measure  of  it, 
or  to  divide  or  measure  it.  A  whole  number  which  divides 
two  or  more  whole  numbers  exactly  is  said  to  be  a  co7nmon 
divisor  or  commmi  measure  of  them. 

Divisor  or  Measure.    Common  Divisor  or  Measure. 


CS  ELEMENTARY    ALGEBllA. 

In  Algebra,  an  expression  wliicli  divides  another  expres- 
sion exactly  is  said  to  be  a  divisor  or  77ieasicre  of  it.  An  ex- 
pression which  divides  two  or  more  expressions  exactly  is 
said  to  be  a  common  measure  or  common  divisor  of  them. 

ISToTE. — The  English  use  the  word  measure  ;  the  French,  the  words 
divisor  and  divide  in  the  same  sense.  We  shall  use  the  latter,  as  they 
have  been  generally  adopted  in  this  country. 

-  89.  In  Arithmetic,  the  greatest  common  divisor  of  two  or 
more  whole  numbers  is  the  greatest  whole  number  which 
will  divide  them  all.  The  expression  Greatest  Common  Di- 
visor, in  Algebra,  must  be  understood  as  applying,  not  to 
the  numericcd  magnitude  of  the  quantity,  but  to  its  dimen- 
sions only;  on  which  account  it  is  sometimes  called  the 
Highest  Common  Divisor, 

The  expression  Greatest  Common  Divisor  is,  however,  re- 
tained in  accordance  with  established  usage,  and  we  shall 
use  the  letters  g.c.d.  for  shortness,  to  indicate  it. 

90.  The  following  is  the  rule  for  finding  the  g.c.d.  of  mo- 
nomials : 

Find  ly  Aritlimetic  the  g.c.d.  of  the  Numerical  Coeffi- 
cients ;  after  this  number  put  every  letter  which  is  common 
to  cdl  the  monomials,  giving  to  each  letter  respectively  the  least 
exponent  which  it  has  in  the  monomials, 

91.  T'or  example  :  required  the  g.c.d.  of  IQa^Vc  and  20a^b\L 
Here  the  numerical  coefficients  are  16  and  20,  and  their  g.c.d. 
is  4.  The  letters  common  to  both  the  expressions  are  a  and 
I) ;  the  least  index  of  a  is  3,  and  the  least-  index  of  b  is  2. 
Thus  we  obtain  4a^^^  as  the  required  g.c.d. 

Again :  required  the  g.c.d.  of  Sa^Fc^'x^yz^,  12a'^l?cx^y%  and 
16a^cVy\  Here  the  numerical  coefficients  are  8, 12,  and  16  ; 
and  their  g.c.d.  is  4.     The  letters  common  to  all  the  expres- 

Rule  for  the  Greatest  Common  Divisor. 


GREATEST   COMMON    DIVISOR.  69 

sions  are  a^  c,  x,  and  y ;  and  their  least  indices  are  respect- 
ively 2,  1,  2,  and.  1.     Thus  we  obtain  ^a^cx'y  as  the  required 

G.C.D. 

92.  Tlie  folloAving  statement  gives  the  best  practical  idea 
of  what  is  meant  by  the  term  greatest  common  divisor,  in 
Algebra,  as  it  shows  the  sense  of  the  word  greatest  lic^e. 
When  ttuo  or  more  exjjressions  are  divided  dy  their  greatest 
common  divisor,  the  quotients  have  no  common  divisor. 

Take  the  first  example  of  Art.  91,  and  divide  the  expres- 
sions by  their  g.c.d.  ;  the  quotients  are  4ac  and.  bhd,  and 
these  quotients  have  no  common  measure. 

Again,  take  the  second  example  of  Art.  91,  and  divide  the 
expressions  by  their  g.c.d.  ;  the  quotients  are  2Fcx^z^,  Zc^ly''^ 
and  4«cy,  and  these  quotients  have  no  common  measure. 

93.  The  idea  which  is  supplied  by  the  preceding  Article, 
with  the  aid  of  the  Chapter  on  Factors,  will  enable  the 
student  to  determine  in  many  cases  the  g.c.d.  of  compound 
expressions.  For  example:  required  the  g.c.d.  of  AiC^(ci-\-l)Y 
and  ^ab  {a^  —  lf).  Here  2a  is  the  g.c.d.  of  the  factors  ^a"^  and 
Q>ah ;  and  6^+^  is  a  factor  of  {a-\-hy  and  of  a^—lf,  and  is  the 
only  common  factor.  The  product  2a{a  +  I))  is  then  the 
G.C.D.  of  the  given  expressions. 

The  rule  in  this  case  is  similar  to  that  given  in  Art.  90 : 
Put  doto7i  every  factor  common  to  cdl  the  expressiojis,  giving 
to  each  factor  respectively  the  least  exponent  luhich  it  has  in 
the  exp)ressions.    The  product  of  these  factors  iviU  be  the  Great- 
est  Common  Divisor  of  the  expressions, 

Ex.  1.  The  G.C.D.  of  15.?;'  and  l^x\  is  ^xJ", 

Ex.  2.  The  g.c.d.  of  SG.r'^V  and  A:%x'y'z\  is  12x\fz\ 

Ex.  3.  The  g.c.d.  of  SoaWx't/  and  ^^a^Vx'y\  is  Ha'b'xy. 

Ex.  4.  The  G.C.D.  of  oax''  —  2a"x  and  a'^x^—dabx,  is  ax. 

The  G.C.D.  of  Compound  Expressions. 


70  ELEMENTARY    ALGEBRA. 

Ex.  5.  The  G.c.T>.of  6x'y-12xY-\'3xy' 

and  4:ax'^  -}-  4:axy  +  4:a^x,  is  x. 
Ex.  6.  The  g.c.d.  of  6aV  {a'-x')  and  4^\r  (a+^)\ 

is  2a'^x  {a  +  x), 
Ex.  7.  The  G.C.D.  of  a\a'x' -Sax' +  2x')  and  x^a'-iaV)^ 
that   is,  of   0^x^(0^— 3ax  +  2x'^)    or    aV(a— 2a;)  (a— a;)  and 
aV(a'--4a;''),  is  aV(a~ 2:c). 

Examples — 15. 
Find  by  inspection  the  g.c.d.  of 

1.  Ax'  {a+xy  and  10  {a'x-x'y. 

2.  x'  {a'-xy  and  {a'x+axy. 

3.  {a'b-aby  and  a^  (a^-5^)^ 

4.  6  (:^;'-l)  and  8  (2;^-3.r+2). 

5.  {x^+xy  and  a;'  (:r'-2:~2). 

6.  4  (ic'+a')  and  6  {x'-2ax-3a''). 

7.  a'  (^^+12^+11)  and  a'x'-lla'x-12a\ 

8.  9  (aV-4)  and  12  (^V  +  4(^:?;+4. 

9.  :?;'  — 9a;+14  and  a:'— lla;4-28. 

10.  a;''  +  8a;+15  and  ic'  +  9:r+20. 

11.  a;'+2:r-120  and  a;'  — 2a;-80. 

12.  4  (x'-x-hl)  and  3  (a;'+a;'  +  l). 

13.  x'-xy-ny'  and  a;'+5a:?/+6^'. 

94.    The  G.C.D.  of  two  polynomials  cannot  be  generally 
fonnd,  however,  thus  by  inspection.     Hence,  for  more  com- 
plex examples  it  is  necessary  to  adopt  another  method — the    . 
same  given  in  Arithmetic  for  two  numbers. 


GREATEST   COMMOIS^   DIViSCR.  71 

95.  Let  there  be  given  then  two  algebraic  quantities  of 
which  it  is  required  to  find  the  g.c.d. 

EuLE. — Arrange  the  quantities  according  to  poivers  of 
some  common  letter,  and  divide  the  one  of  higher  dimen- 
sions  dy  the  other ;  or,  if  the  highest  exponent  happen  to  he 
the  same  in  each,  taJce  either  of  them  for  dividend.  Take 
now,  as  in  Arithmetic,  the  remainder  after  this  division  for 
divisor,  and  the  preceding  divisor  for  dividend,  and  so  on  until 
there  is  no  remainder ;  then  the  last  divisor  tvill  ie  the  g.c.d. 
of  the  tiuo  given  quantities, 

Ex.    Find  the  g.c.d.  of  2;' -  7^ -f  10  and  4x'~25:z;'+20:r+25 

a;'-7ii;+10)4a;'-25a;'  +  20:2;-f25(4:?;+3 
4a;^-28:^:''  +  40:y 

3.^'- 20a; +25 

x-h)x^'-nx^-\^(x-% 
x^  —  hx 

-2^+10 

—  2a;+10       Ans.   ic— 5. 


Examples — 16. 

Find  the  g.c.d. 

1.  Of  32)'+  ^-2  and  3.T'+4a;--4. 

2.  Of  6:?;'  +  7a;-3  and  12a;'  +  16rr-3. 

3.  Of  9a;'-~2o  and  9a;''  +  3:r-20. 

4.  Of  8a;'  +  14.T-15  and  8a;' + 30a:' +  13a;- 30. 

5.  Of  4a;'+3a;-10  and  4a;' +  7a;' -3a; -15. 

6.  Of  2a;* +a;'- 20a;' -7a; +  24  and  2a;* + 3a;' -13a;' -7a; +15. 

Rule  for  the  g.c.d.   of  Polynomials. 


72  ELEME^s'TARY    ALGEBRA. 

96.  In  order  to  prove  the  Eule  above  given,  it  will  bt* 
necessary  to  show  first  the  truth  of  the  following  statement 

If  a  quantity  c  he  a  common  divisor  of  a  and  b,  it  Krill  also 
divide  the  sum  or  difference  of  any  multiioles  of  a  and  b,  as 
ma  ±  nb. 

For  let  c  be  contained  p  times  in  a  and  q  times  in  I ;  then 
a—pc,  h—qc,  and  ma  ±  nh  —  mi^c  -ihnqG—  {mp  ±  ncfjc ;  hence 
c  is  contained  mp  ±  nq  times  in  ma  ±  nh,  and  therefore  c 
divides  am^  ±  nh. 

Thus,  since  6  will  divide  12  and  18  without  remainder,  it  will  also 
divide  any  number  such  as  7x12  +  5x18,  11x12  —  3x18,  12  (or  1x12) 
+  7x18,  5x12—18,  &c.,  /.  6.,  any  number  found  by  adding  or  subtract- 
ing any  multiples  of  12  and  18. 

97.  To  prove  the  Rule  for  finding  the  Greatest  Common  Di- 
visor of  tioo  quantities. 

First,  let  the  two  given  quantities,  denoted  by  a  and  5, 
have  neither  of  them  any  monomial  factor. 

Let  a  be  that  which  is  not  of  lower  dimensions  than  the 
other;  and  suppose  a  divided  by  Z>,  with  quotient  j!9  and  re- 
mainder c\  bhj  c,  with  quotient  q  and  remainder  d,  &c. 

b)  a  {p  546)  672  (1 
2)b_  546 

c)l){q  126)  546  (4 

'  qc_  504 

d)c{r  42)126(3 

rd  126 

Then,  by  (96),  all  the  common  divisors  of  a  and  Z>.are  also 
divisors  of  a—ph  or  c,  and  are  therefore  common  divisors  oT 
J)  and  c;  and  conversely,  all  the  common  divisors  oib  and  o 
are  also  divisors  oi p)b-{-c  or  a,  and  are  therefore  common  di- 
visors of  a  and  h.  Hence  it  is  plain  that  b  and  c  have  pre- 
cisely the  same  common  divisors  as  a  and  b. 

Troof  of  tlie  Rule  for  the  g.c.d.  of  Polynomials. 


GREATEST   COMMOI^   DIVISOR.  ^73 

111  like  manner  it  may  be  shown  that  c  and  d  have  the 
same  common  divisors  as  h  and  c,  and  therefore  the  same  as 
a  and  h. 

And  so  we  might  proceed  if  there  were  more  remainders, 
the  quantities*  a,  d,  c,  d,  &c.  getting  lower  and  lower,  yet  still 
being  such  that  a  and  1),  h  and  c,  c  and  d,  &c.  have  the  same 
common  divisors. 

But,  if  d  divides  c  without  remainder,  then  d  is  itself  the 
greatest  quantity  that  divides  both  c  and  d\  that  is,  d  is  the 
greatest  of  the  common  divisors  of  c  and  d,  and  therefore  is 
the  Greatest  Common  Divisor  of  a  and  h. 

Thus,  in  the  numerical  example,  the  common  divisors  of  546  and  672 
ax-e  precisely  the  same  as  those  of  126  and  546,  and  these  again  are  the 
same  as  those  of  42  and  126 ;  but  42  is  the  g.c.d.  of  42  and  126,  and  is 
therefore  the  g.c.d.  of  126  and  546,  and  also  of  546  and  672. 

(See  Venable's  Arithmetic,  Arts.  82,  83.) 

98.  If  the  original  expressions  contain  a  common  fiictor  F, 
which  is  obvious  on  inspection,  then  this  factor  F  will  be  a 
factoi  of  the  g.c.d.  We  strike  it  out  from  both  the  quanti- 
ties and  apply  the  rule  to  the  resulting  quantities.  The 
G.c.D.  ihus  found  must  be  multiplied  by  F  to  get  the  g.c.d. 
of  the  original  quantities. 

99.  If  either  of  the  quantities  contain  a  factor  which  is 
obviously  not  a  factor  of  the  other,  this  must  be  struck  out, 
and  the  g.o.d.  of  the  resulting  quantities  is  the  g.c.d.  of  the 
original  quantities. 

So,  whenever  we  take  a  Eemainder  for  a  Divisor  in  apply- 
ing the  rule,  we  may  strike  out  any  simple  factor  it  may 
contain. 

100.  Again,  if  after  having  thus  prepared  the  divisor,  at 
any  step  of  the  process  we  find  that  the  first  term  of  the  divi- 
dend is  not  exactly  divisible  by  the  first  of  the  divisor,  then, 

other  features  of  the  process  of  finding  the  g.c.d. 

4 


74  ELEMEXTAET    ALGEBRA. 

in  order  to  avoid  fractions  in  the  quotient,  we  may  multiply 
the  whole  dividend  by  such  a  simple  factor  as  will  make  its 
first  term  so  divisible. 

In  general,  we  may  divide  the  divisor  hy  any  cxjjression 
wliicli  has  no  factor  commo7ito  the  hoo  quantities  luhose  G.C.D, 
we  are  seeking ;  or  tue  may  multiply  the  dividend  hj  any  ex- 
pression luhich  has  no  factor  common  to  the  divisor. 

Ex.  Find  the  g.c.d.  of 

^x'-^x^'-^l^x'-^x'  +  ^x  and  dx'-Qx'^-^x. 

Here,  striking  out  of  the  first  the  factor  2x  (which  is  com- 
mon to  all  its  terms),  and  of  the  second  the  factor  3:r,  we  re- 
duce the  quantities  to  x^—4:X^-\-Gx^—4,x+l  and  ^^— 2^?;^+!; 
but  as  2x  and  3x  have  themselves  a  common  factor  x,  it  is 
plain  that  the  original  quantities  have  a  common  factor  x, 
which  these  latter  quantities  have  not;  hence  the  g.c.d.  of 
these,  when  found,  must  be  multiplied  by  x  to  produce  that 
of  the  given  quantities. 

x''-2x'  +  l)x'-4:x'+6x'-4.x+l{l 


-ix\- 

-  4a;' +  8a;' -4a; 

x' 

-3a;+l 

x"- 

-2x+l)x*- 

-3a;' 

+  l(a;'-l-3a;  +  l 

X*- 

-2a;' 

^-x' 

%x'- 

^3a;'  +  l 

3a;'- 

-4a;'+3a; 
a;'— 2a;  +  l 
a;'-3a;+l 

In  this  example,  the  first  remainder  is  reduced  by  dividing 
it  by  —4a:;  and,  the  g.c.d.  of  these  two  quantities  being  x^— 
2a;  4-1,  that  of  the  two  given  q?  antities  will  be  x  (.t'  — 2a;+l) 
or  x^—2x^  +  x. 


GREATEST   OOMMOK    DIVISOll.  75 

Ex.  Find  the  g.c.d.  of 

Qx'y  +  ^xif  —  ^y^  and  ^x^-^4cX^y—4:X7f, 

Stripping  them  of  their  simple  factors  2y  and  4a;  ( ind 
noting  that  these  contain  the  common  factor  2),  we  }  ave 
^x^-{-2xy—y'^  and  ^x^  +  xy—y"^,  and  proceed  with  these  qi  en- 
tities as  follows : 

2 


2x'  +  xy'-y')^x^  +  4.xy—2y\^ 
6^jy3xy-3f 
y)xy+y^ 

x  +  y)2x^  +  xy—y^{2x—y 
2x^  +  2xy 

—xy—y"^ 


The  G.C.D.  then  will  be  2  (a;  +  y) ;  it  being  plain  that  the 
G.C.D.  of  2{^x'-[-2xy'-y^)  and  2x'-\-xy—y''  will  be  the  same 
as  that  of  dx^-^^xy—y"^  and  ^x^  +  xy—y"^,  because  the  2  intro- 
duced into  the  first  is  no  factor  of  the  second  quantity. 

Examples — 17. 
Find  the  g.c.d. 

1.  Oilox''-x-Q  and  ^x'-Zx-^, 

2.  Of  6a;'-a;-2and21a;'-26a;'  +  8a;. 

3.  Of  2a;' +  6a;' +  6a:  4-2  and  6a;'' +  6a;' -6a; -6. 

4.  Of  2/- 10?/' +  12?/  and  3?/*- 15/ +  24?/' -24. 

5.  Of  x'  -  6ax'  +  12^'a;— 8^'  and  x'  -  4a'.a;'. 

6.  Of  2x'  +  10a;'  +  14a;  +  6  and  x'  +  x'-}-  7x  +  39. 

7.  Of  3a;^  +  3a;'-15a;  +  9  and  e3a;^  +  3a;^-21a;'-9aj. 


76  ELEMEJS'TARY    ALGEBRA. 

9.  Oi2a'  +  a'h-4.a'h''-Uh'  and  W ■\-a'h-2a''lf +  ab\ 

10.  OiM'-\-16a'h-'da'h''-lba'h' 

and  10a'-^0a'h-lQa''¥-\-'d0al\ 

11.  O^x'-^x'y  +  ^xy^-y'  and  x'-2x'y^^x''y''-'^xif  +  y\ 

12.  Of  a;'+6a;'  +  ll^'+42;-4and2;'+2a;'-5:?;'-12a;-4. 

XIII.    Least  Common  Multiple. 

101.  When  one  Q\ndin.i\ij  contains  another  as  a  divisor  with- 
out remainder,  it  is  said  to  be  a  multiple  of  it ;  and  a  common 
multiple  of  two  or  more  quantities  is  one  that  contains  each 
of  them  without  remainder. 

Thus,  Qx^y  is  a  common  multiple  of  2a^^,  Zxy,  6.2;^,  &c.,  and  any  quan- 
tity is  a  multiple  of  any  of  its  divisors. 

102.  The  Least  Common  Multiple  of  two  or  more  alge- 
braic expressions  is  a  term  not  appropriate  if  we  use  it  in  the 
arithmetical  sense.  We  must  understand  it  to  mean  the 
quantity  of  lowest  dimensions  which  is  exactly  divisible  by 
these  expressions.  As  in  Arithmetic,  we  will  use  the  letters 
L.C.M.  for  shortness. 

103.  To  find  the  l.c.m.  of  simple  expfressions  or  monomials : 

Fi7id  by  Arithmetic  the  l.c.m.  of  the  nwnerical  coefficients  ; 
after  this  number  put  every  letter  luhich  occurs  in  the  expres- 
sions, and  give  to  each  letter  respectively  the  greatest  exponent 
lohich  it  has  in  the  expressions. 

Ex.  Find  the  l.c.m.  of  IQa^'bc  and  20aWd.  Here  the  l.c.m. 
of  16  and  20  is  80.  The  letters  which  occur  in  the  expres- 
sions are  a,  b,  c,  d;  and  their  greatest  exponents  are  4,  3,  1, 
and  1.     The  required  l.c.m.  is,  therefore,  80a*b^cd. 

Meaning  of  Least  Common  Multiple  in  Algebra.    Rule  for  l.c.m.  of  Monomials, 


LEAST   COMMO]Sr   MULTIPLE.  77 

Ex.  Find  the  l.c.m.  of  ^a^h'c'x'y^,  l^a'hcxY  and  IMc'x'if, 

Here  the  L.  c.  m.  of  the  numerical  coefficients  is  48.  The 
letters  which  occur  are  a,  h,  c,  x,  y,  and  z]  and  their  greatest 
exponents  are,  respectively,  4,  3,  3,  5,  4,  and  3.  Thus  we 
obtain  ^^a^lfc'x^y^z'  as  the  required  l.  c.  m. 

104.  We  shall  now  show  how  to  find  the  l.c.m.  of  two 
compound  expressions  or  polynomials. 

Let  a  and  h  represent  the  two  quantities,  d  their  G.  c.  D. : 
and  let  a~pd,  h—qd,  so  that  jt?  and  q  will  have  no  common 
factor.  Then  the  least  quantity  which  contains  p  and  q  will 
be  pq,  and  therefore  the  least  quantity  which  contains  pd 
and  qd  will  \}q pqd,  which  is  consequently  the  l.c.m.  required 
of  a  and  d, 

Bince pqd=- — --^=— — -,  it  appears  that  the   l.c.m  of 

a  and  b  may  be  found  by  dividing  their  product  iy  their 
G.c.D. ;  or,  which  is  more  simple  in  practice,  by  dividing 
either  of  them  by  their  g.c.d.,  a7id  multiply i7ig  the  quotient  by 
the  other. 

For  example:  required  the  l.c.m.  of  2;"^— 4x+3 
and42;'-9:c'^-152;+18. 

The  G.c.D.  is  x—'d]  see  Art.  95.  Divide  x'—4zX-\-^  by 
x—o\  the  quotient  is  x—1.  Therefore  the  l.c.m.  is 
{x—l){4:X^  —  ^x'^—lhx-\-l'^)\  and  this  gives,  by  multiplying 
out,  4.x'-l'dx'-Qx''-\-?,dx-lS, 

It  is,  however,  often  convenient  to  have  the  l.c.m.  ex- 
pressed in  factors,  rather  than  multiplied  out.  "We  know 
that  the  g.c.d.,  which  is  ^—3,  will  divide  the  expression 
-ix^'—^x^  —  lbx  +  l^]  by  division  we  obtain  the  quotient. 
Hence  the  l.c.m.  is 

{x-Z){x-l){4.x^-dx-Q). 

105.  The  principle  of  the  rule  in  Art.  103,  with  the  aid  of 

L.C.M..  by  Inspection. 


78  ELEME]S"TAEY    ALGEBRA. 

tlie  Chapter  on  Factors,  will  enable  ns  in  many  cases  to  de- 
termine, by  inspection,  the  L.C.M.  of  polynomial  expressions; 
as  we  liaye  only  to  set  doiun  the  factors  luMcli  compose  them, 
each  affected  with  the  highest  exponent  ivhich  it  has  in  the  ex- 
pressions ;  and  the  product  of  these  is  the  Jj.cm,  required, 

Ex.  1.  Find  the  l.c.m.  of  ^hx,  Qahxy,  dacx. 

Here  the  factors  are  2bXf  day,  c ;  and  the  l.c.m.  is  6abcxy, 

Ex.  2.  Find  the  l.c.m.  of  2a\a  \-x),  4:ax  {a—x),  6x''{a+x), 
Here  the  l.c.m.  of  the  simple  factors  is  12aV,  and  that  of 

the  coniponnd  factors  is  a^—x'^;  therefore  the  l.c.m.  required 

is  12a'x'{a'-x'), 

106.  Every  common  multiple  of  two  quantities  A  and  B,  is  a 
multiple  of  their  l.c.m.  For,  let  M  denote  the  l.c.m.  of  A  and 
B,  and  ]N"  some  other  multiple.  Suppose  that  if  possible  when 
!N"  is  divided  by  M  there  is  a  remainder  K ;  let  ^^  denote  the 
quotient.     Then, 

Isr=M^  +  E,  and  E=K-M^. 
J^ow  since  A  and  B  divide  both  M  and  IST,  they  divide,  also^ 
IST—Mg  or  K  Therefore  R,  which  from  the  nature  of  di- 
vision is  of  loioer  dimensions  than  M,  is  a  multiple  of  A  and 
B  less  than  the  l.c.m.  This  is  absurd.  Therefore  there  can 
be  no  remaiiider  E.    That  is,  N  is  a  multiple  of  M. 

107.  Hence  to  find  the  l.c.m.  of  several  expressions,  we  may 
find  the  l.c.m.  of  two  of  them ;  then  find  the  l.c.m.  of  this 
first  L.C.M.  and  the  third  expression,  and  so  on. 

Examples — 18. 

Find  the  l.c.m. 
1.  Of   ^a%c  and   Qali^c',    of   ^x^y  and   12xy^  \     of   axy 
and  a{xy—y'^) ;  of  ab  +  ad  and  ab—ad, 

L.C.M.  of  more  than  two  expressions. 


(   VNIVERSITY 

^LE.  79 

2.  Of  ^a%  l^a%  and  12a'Jf',   of  a%  ba%  lQa'h\  10d'b% 

bah\   and  h" )    of   ^x",  6ax,   8a^  36x%  dax",  60a% 
and  24a'. 

3.  Of  2(a  +  Z>)  and  S{a'-b');   of  4(a'-a)  and  6{a'  +  a)\ 

of  6(:^;'  +  :^•^),  S{xy-y'),  and  10(cz;'^-/). 

4.  Of  4(a'-aZ»^),  12(aZ)^  +  Z^'),  S{a'-a'b); 

and  of  6{x''y  +  xy^),  9{x^—xy''),  4z{y^+xy''). 

5.  Of  a;'-3a;-4,  :i;'-a;-I2. 

6.  Of  i^H5.^•H7^^-2,  ic^  +  6cc  +  8. 

7.  Of  12a;^  +  5^-3,  Qx'+x'-x, 

8.  Of  a;*-' -  6a;' + 11:^-6,2;'- 92;' +  26a;-24. 

9.  Of  x'-7x-6,  a;''+8a;'  +  17a;  +  10. 

10.  Of  x'  +  x'  +  2x'  +  x  +  l,  x'-l. 

11.  Of  4a^Z>'c,  6«^V,  -^Sa'M. 

12.  Of  8(a'-^>'),  12(a  +  ^)',  20{a-'^)\ 

13.  Of  4(a  +  ^),  6(a'-^'),  8{a'  +  ¥), 

14.  Of  15(a'Z>-a^'),  21(a^-a&'),  36{ab'  +  b'). 

15.  Of  i?;'-l,  c?;^+l,  :?;^-l. 

IG.  Of  x'-l,  x'+l,  x'  +  l,  x'-l. 

17.  Of  x'-l,  x'  +  l,  x'-l,  x'  +  l. 

18.  Of  a;'  +  3a;  +  2,  x'  +  4:X-i-d,  x'  +  5x  +  6. 


80  ELEMENTARY     ALGEBEA. 


XIV.    Feactions. 

108.  Algebraical  fractions  are  for  the  most  part  precisely 
similar,  both  in  their  nature  and  treatment,  to  common 
arithmetical  fractions.  Hence,  the  student  will  find  the 
rules  and  demonstrations  in  the  chapters  on  Fractions  are 
little  more  than  a  repetition  of  those  with  which  he  is 
already  familiar  in  Arithmetic. 

109.  The  expression  -7-,  we  have  agreed  shall  denote  that 

a  is  divided  by  h.  We  now  say  -j-  means  that  the  unit  is  di- 
vided into  1)  equal  parts,  a  of  w^hich  are  taken :  a  is  called 
the  numerator,  and  h  the  denominator,  and  the  expression  -j- 

is  a  fraction.  (We  shall  show,  as  in  Arithmetic,  that  a  frac- 
tion does  also  express  the  quotient  of  the  numerator  divided 
by  the  denominator.) 

Every  integral  quantity  may  be  considered  as  a  fraction 
whose  denominator  is  1.    Thus, 

a—-,  o  +  c=—-— 

110.  llo  multiply  2b  fraction  by  an  integer:  Either  multi- 
ply the  numerator,  or  divide  the  denominator  hy  the  integer ; 
and  conversely,  to  divide  a  fraction  by  an  integer :  Either 
divide  the  numerator,  or  multiply  the  deoiominator  ly.  the  in- 
teger. 

Thus,  -Xx=—  ;  for  in  each  of  the  fractions  ^,  -^,  the 
00  ho 

unit  is  divided  into  h  equal  parts,  and  x  times  as  many  of 

them  are  taken  in  the  latter  as  in  the  former ;  hence  the  lat- 


Algebraic  Fractions.    To  multiply  or  divide  a  fraction  by  an  integer. 


FRACTIONS.  81 

ter  fraction  is  x  times  the  former,  that  is,  —  =— -Xcc;  and, 
ax 


by  similar  reasonino:,  -^-^ccr=- 


A^ain,  —-^x—-i-\  for  in  each  of  the  fractions  -r-,  t-»  the 
^         b  bx'  b    bx^ 

same  number  of  parts  is  taken,  but  each  of  the  parts  in  the 
latter  is  -th  of  each  in  the  former,  since  the  unit  in  the  lat- 

X 

ter  case  is  divided  into  x  times  as  many  parts  as  in   the 
former ;  hence  the  latter  fraction  is  -th  part  of  the  former, 

that  is,  -—-—--^x\  and,  similarly,  -—Xx=-, 

111.  If  any  quantity  be  both  multiplied  and  divided  by  the 
same  quantity,  its  value  will,  of  course,  remain  unaltered. 
Hence,  if  the  numerator  and  denominator  of  a  fraction  be 
both  multiplied  or  divided  by  the  same  quantity,  its  value  loill 
remain  unaltered, 

^,  a     ax     a^      ,,  ,    a3       a     ac     ^ 

ilius,    -T- =-7-=-T  =<!^c.,  and    -— -  =  -— — =&c. 
b      bx     ab  a  be     c      c 

This  result  is  of  great  importance,  and  many  of  the  op- 
erations in  Fractions  depend  on  it. 

112.  To  reduce  an  integer  to  a  fraction  with  a  given  de- 
nominator: Multiply  it  by  the  given  de^iominator,  and  the 
product  will  be  the  numerator  of  the  required  fraction. 

The  truth  of  this  is  evident  from  (110). 

Thus,  a  expressed  as  a  fraction  with  denominator  a;  is  — ; 

.,,    ,  •     .      7         .    ctb—ac 

or  with  denominator  /9— c,  is  -^ . 

b  —  c 

113.  Since  a  —  —,  and  therefore  a  divided  ]dy  b  =~-^b=--r 

1  -^  1  b 

(109),  it  follows,  as  stated  in  (108),  that  a  fraction  represents 

To  reduce  an  integer  to  a  fraction  with  a  given  denominator. 

4« 


82  ELEMENTARY    ALGEBRA. 

the  quotient  of  tlie  numerator  by  the  denominator.  In  fact 
we  get  -y-th  of  a  units  (or  a-^h)  by  taking  — th  of  each  of 
the  a  units,  and  this  is  the  same  as  a  such  parts  of  one  unit 
which  we  haye  expressed  (108)  by  -^. 

1  3 

Thus,  in  Arithmetic,  —  of  $3  is  the  same  as  —  of  $1. 

III.  The  demonstrations  giyen  in  the  preceding  Articles 

are  based  on  the  assumption  that  eyery  letter  denotes  some 

2)ositive  luliole  numler.    By  the  Rule  of  the  Sigjis  established 

in  Multiplication  and  Diyision,  we  haye  the  following : 

c^'  (^      CIO    ^  ...  ^    ^  ^         a      —a 

bmce   —  =  — ,    by  puttmg    —1   tor  c  we  haye  —  =  — -, 

Hence,  we  may  change  the  signs  of  all  the  terms  in  loth  the 
numerator  and  denominator  of  a  fraction  without  altering 
its  value, 

^  x^—^ax—a^  .    .,     ^.    ,     .,,    a^-\-2ax—x^ 

Jix.        — ^—  is  identical  with  ^ — . 

dax—x  X  —3ax 

^      ^         a        ^  —a        a 
So  also, -^=+-^-=-^5 

—b  ~       h  ~  h' 

In  like  manner,  by  assuming  that  -^-X^?  is  equal  to  -j-, 

whateyer  be  the  sign  of  c,  Ave  obtain  such  results  as  the  fol- 
lowing : 

115.  If  the  numerator  of  a  fraction  be  of  lower  dimen- 
sions than  the  denominator,  the  fraction  may  be  considered 
in  the  light  of  a  proper  fraction  in  Arithmetic  :    if  the  nu- 

Changing  the  signs  of  tlie  terms. 


FRACTIOI^fo.  83 

merator  be  of  liiglier  dwiensions  than  the  denominator,  it 
may  be  considered  in  the  light  of  an  improper  fraction, 
which  (Art.  112)  in  Algebra,  as  in  Arithmetic,  may  be  ex- 
pressed as  a  mixed  quantity  by  the  rule : 

Divide  the  nunierator  hy  the  denominator  as  far-as  the  di- 
vision  is  possible,  and  annex  to  the  quotient  a  fraction  having 
the  remainder  for  numerator  and  the  divisor  for  the  denom- 
inator. 


Thus, 


24a     „    ,  M 

-  —  aJr^O- 


= —  —  M.  -r  -vt/ 7. 


x'-dx-V^  x'-dx  +  4:'  x'-3x-\-4: 

This  last  step  the  student  should  particularly  notice,  as 
an  example  of  the  use  of  brackets,  namely : 

+  (-x  +  2)  =  -{x-2). 

Examples — 19. 

Express  the  following  fractions  as  mixed  quantities. 

25x  36ac+4.c.  Sa^'  +  Sb, 

7  9  4flj 

,    nx'-hy  ^    a;'4-3.'c+2  ^    2x'-6x-l 

6x  x+3  x-3 

^    x'-\-ax'-3a:'x-3a'  ^    x'-2x'' 


x—2a  '  '  x^  —  x  +  1' 

To  reduce  improper  fractions  to  whole  or  mixed  quantities. 


84:  ELEMEKTAEY    ALGEBRA. 

x—1  x  +  1 

Multiply 

■,•,    4a'       ^,  ,„    8(0.' +b'')  ,     „,      ,, 

11.  g^  by  35.  12.  A^_^by3(a-5). 

Diyide 

15.  -^—  by  2a;.  16. —r-  by  3a— 2^. 

18.  ^^hjx^^x+1. 


XV.    Eeduction  of  Feactions. 

116.  The  result  contained  in  Art.  110  will  now  be  applied 
to  the  reduction  of  a  fraction  to  its  lowest  terms,  and  the  re- 
duction of  fractions  to  a  common  denominator. 

117.  Eule  for  reducing  a  fraction  to  its  lowest  terms: 

Divide  the  numerator  and  denominator  of  the  fraction  hy 
their  Greatest  Common  Divisor, 

For  example :  reduce  ^7—^,7-^  to  its  lowest  terms. 

%i         —  ■ ■ 

To  reduce  a  fraction  to  its  lowest  terms. 


REDUCTIO^q^   OF   FRACTIONS.  85 

Dividing  both  numerator  and  denominator  by  AiCi^h'^,  which 

4i:ClC 

is  their  g.c.d.,  we  obtain  for  the  required  result  -^j-.     That 

is,  -^j-j  is  equal  to  KTrrfJl  {\^^)y  ^^^  i^  is  expressed  in  a  more 
simple  form. 

Ao:ain :  reduce  -j—, — —  ,\^ — — -r  to  its  lowest  terms. 

Dividing  both  numerator  and  denominator  by  x—'d,  which 

x—1 
is  their  g.c.d,,  we  obtain  for  the  required  result  j-ti-q^ — F 

^X  ~\~  ijX  —  o. 

118.  In  many  examples  we  may  apply  the  results  ol  the 
chapter  on  Factors,  and  strike  out  the  common  factors 
from  the  numerator  and  denominator  without  using  the 
rule  for  finding  the  g.c.d.  ;  or  rather,  we  may  ly  mere  in- 
sjpedion  find  the  g.c.d.  (Art.  93),  and  strike  it  out  from  the 
7mmerator  and  denominator. 


Ex.  1. 


o^x^y'^      _      a^x^y^     _^  axy 


a^xy  +  axy"^     axy  {a  +  y)     a  +  y 


^  a^-{-x^_{a-\-x)  (a^—ax+x^)     a^—ax^-x^ 

a  —X  {a+x)  {a—x)  a—x 

Ex    3    t±^±lJ^^li^J^J^±l 
a;'  +  5:^  +  6     (2:  +  3)  {x±2)'~x-{-'-Z' 

x'^x''-\-^x-^_{x-l)  {x'-\-2x^b)_x''  +  '^x-\-5 
^^'     •      x'-^x-\-'d     ~~     {x-1)  {x-d)      ~     a;-3"~"* 

-p,      ^    {a—hy—c^_{a—h-\-c){a—h  —  c)_a  —  h-^c 
a'—{h-\-cy~\a-^b  +  c){a—b  —  c)~a  +  b-\-c 

119.  Kule  for  reducing  fractions  to  a  common  denominator : 
Multiply  the  numerator  of  each  fraction  ly  all  the  denom* 

To  reduce  a  fraction  to  its  lowest  terms  by  Inspection.    To  reduce  fractions 
to  a  Common  Denominator. 


86  ELEMENTARY    ALGEBEA. 

inators  exceipt  its  own  for  the  numerator  corrcsjjonding  to  that 
fraction ;  and  muUijply  all  the  denominators  together  for  a 
common  denominator. 

The  truth  of  this  rule  is  eyident;  since  the  numerator  and 
denon\inator  of  each  fraction  are  hoth  multiplied  by  the  same 
quantities  (viz.  the  denominators  of  the  other  fractions),  its 
Taluo  will  not  be  altered,  though  all  the  fractions  will  now 
appear  with  the  same  denominator. 

For  example :  reduce  -r  ?  -^  ?  and  -r ,  to  a  common  denom- 

mator. 

a_adf    c  _chf    e__ebd 
iThdf  2~dbf'f~fkV 

Thus,  T-j-^,  -7^,  and  -7^-7  are  fractions  of  the  same  value, 
odf    dof  fbd 

respectiyely,  as  -tj-^?  and  —  ,  and  they  haye  the   common 

denominator  hdf. 

We  often  wish  to  reduce  fractions  to  their  lowest  common 
denominator,  which  the  aboye  process  will  not  effect  if  there 
ai-e  any  common  factors  in  the  denominators.  It  is  there- 
fore often  conyenient,  as  in  Arithmetic,  to  use  another  rule. 

120.  Eule  for  reducing  fractions  to  the  lowest  common 
denominator 

Find  the  l.c.m.  of  the  denominators.  This  ivill  he  the  Lowest 
Common  Denominator.  Then  for  the  new  numerators  multi- 
ply the  numerator  of  each  of  the  given  fractions  hy  the  quotient 
ivhich  is  obtained  by  dividiyig  the  l.c.m.  by  its  denominator. 

For  example:  reduce  — ,  — ,  —  ,  to-their  lowest  common 
yz    zx    xy 

denominator.     The  l.c.m.  of  the  denominators  is  xyz ;  and, 

To  reduce  fractions  to  their  LoweBt.  Common  Denominator. 


EEDUCTIO]^   OF   FEACTIOKS.  87 

a      ax      1)      hj      c       cz       ^,  ,  _    ^ 

— = — ,    _— -^     _— — .      'ii-^Q  numerator  and  denom- 

yz     xyz     zx    xyz    xy     xyz 

inator  of  each  fraction  has  been  multiplied  by  the   same  , 
quantity. 

Examples — 20. 

Eeduce  the  following  fractions  to  their  lowest  terms. 

l^a'lfx  ^   a^ah  a^ -\- ah 

ISa'b'y  T^*  '^F^^b' 

^   lOa'x  ^    4:{a+I)y  ^   a'  +  F 


6a'x-15ay''  b(ce^}f)'  d'-V' 

x^-VZx^l  a;'^  +  10a;+21 

22;'  +  ^^'— 15  x^^-{a-^l:))x-\-db 

2.r'  — 192;  +  35*  '    '  x^-\-  {a^c)x-\-aG 

x^-{a^h)x-\-db  3^^3^_36 

•  x^-V{c-a)x-ac  4a;^+33i^;-27* 

\x-\-hY-ia-VcY  x'-^x^\^' 

^^    x^—l^x-^'^X  ^^         :z;'  +  92;  +  20 

Id.  —^ — — — -— .  16.   -^ 


x^-\JoX'-V:  '  x^ ^Ix^ ■\-\^x-\-'S 

x'  +  aV  +  a*  x'^-y 

^'  -  «  .  1«.      ^2«y+l- 


19. 


X  —a 
x^—l      x^-a""      a^  —  V        x^  —  lfx       d^  —  al-Vax—lx 


ax-^a'  x'—aV  d'-b''  x" -{•^hx  +  b'"'  d'-\-ab-\-ax-\-hx 


Eeduce  the  following  fractions  to  their  lowest  common 
denominator. 

3       4        5  ^  9.  rr 

20.  ^,  ^,^.  21. 


^x'  6.^^'  12x''  x  +  V  4.'r4-4'  :r'-l* 


88  ELEMENTARY    ALGEBRA. 

fl^  X           a^          ax 

x—a'  a—x    :a 

a  l          ah 


24.  — 


1  a;  3  4 


25. 


26. 


(x-iy  {x-iy  x+v  {x+iy  x'-i' 

a  a-\-x  ax 


x—a'  x^-^-ax  +  a'"'  x^—a^' 


x^—ax-^a"'  x''-\-ax  +  a'"  ic^  +  aV+a*' 


XVI.    Addition  and  Subtraction  of  Fractions. 

121.  To  add  or  subtract  fractions:  Reduce  tliem  to  a  com^ 
men  denomi7iator  (if  necessary),  a7id  add  or  subtract  the 
numerators  for  a  neiu  numerator  and  retain  tlie  common 
de?iominator, 

Ex.1.   Add  ^  and  ^. 

0  0 

TT       ,1  .     a-\-c-\-a—c    2a 

Here  the  sum  is  ^ =-7-. 

0  0 

Ex.  2.   From take  . 

c  c 

^a-3h    da-'4:I?_4.a-3h-{3a-U)  _4:a-3b-Sa  +  ^b 
c  c     ~~  c  ~~  c 

a  +  b 


To  add  or  subtract  fractions. 


ADDITIO:^"   AISTD   SUBTRACTIOiq"   OF   FRACTI0:N^S.  89 


Ex.  3.    Add  -^  and     "^ 


Here  the  common   denominator  will  be  the  product  of 
a-^h  and  a  —  h,  that  is,  a^^lf, 

c    __c(a—b)^       c    _c{a  +  h) 

Therefore,      ^^  +  -^='A^3^±^^ 
a  +  b     a—b  a  -o" 

_ca—cb-\-ca-\-cb__   2ca 

^      .     _         a  +  b  .  ,     a  —  b 
Ex.  4.   From  7  take 


a—b  a+b' 

The  common  denominator  is  a^—b^, 

a  +  b_{a+b)\    a-b_{a-by 
a-b~  a'-b'''    a-^b"  a'-b'' 

Therefore,     ^"^^    a-^b_{a  +  by-{a-by 


a—b    a  +  b  oi^—V 

_y  +  2ab+F-{a'-2ab  +  b')_  iab 
~  a'-b'  ~a'-r 

Ex.  5.   Add  -4^4-,+  ^  ^"^  ,. 

l-{-x  +  x      l—x-\-x 

{1  +  x)  {l-x^-x'')-\-{l-x)  {l-^-x^-x"")  __         2 
Ex.6.   From   .  "^"^^   ,  take      "^"^ 


l  +  x-\-x'  1  —  x+x^' 

{1  +  x)  (l-x-\-x')-{l-x)  (1  +  ^;  +  :^:') _      2x' 

{l  +  x  +  x'){l-x  +  x')  '~l  +  x'  +  x*' 


90  ELEMENTARY    ALGEBRA. 

122.  AYe  haye  sometimes  to  reduce  a  mixed  quantity  to  a 
fraction ;  this  is  a  simple  case  of  addition  or  subtraction  of 
fractions. 

For  example : 

h      a      h     ac     I     ac-\-h 
c      1      c      c      c        c 


h  __a      h  _ac     h  _ac—h 
c~ 1      c~  c      c~     c 

Hence, 

Multiply  the  entire  part  ly  the  denominator  of  the  frac- 
tion, add  to  or  suMract  from  this  the  numerator  of  the  frac- 
tion, and  place  the  result  over  the  denominator, 

-,     ^  .   2«Z>      a      2«Z>     aia  +  i)      2ab     a^  +  3ab 

Ex.  1.     a-\ 7=-T-H r-7=— r^  H 7= t— • 

a  +  o     1     a+o       a^-0       a-\-o       a  +  o 

x—2         x+3         x-2 


Ex.  2.     x  +  3 


x''-3x  +  4.       1        x'-3x  +  4: 


_{x  +  3)(x'-3x  +  4:)  x-2 

'~        x''-3x-\'4:  x''-3x-\-4. 

_x^'-^x-\-12-{x-2)  _x^—bx  +  12'-x-^2  _x^-^x  +  l^ 

Ex    3     1    ^'+^'-^'_^^^  +  ^'  +  ^' -^'_('^+^)'-^' 
2ah      ~~  2ah  ~'       2ab 

_{a  +  d  +  c)(a  +  h—c) 
"~  2ab 

^       .      ^,        XT,  4.     -,     a^'  +  h^-c^     {a~d  +  c)(I)-a  +  c) 

Ex.4.     Snow  that    1 ^r-^ ==^ -zr^ : 

2ao  2ao 

To  reduce  a  mixed  quantity  to  a  fraction. 


ADDITION    AXD   SUBTRACTIO:^"   OF    FRACTIOXS.  91 

Ex.  5.     Show  that 

a''^]f-c'\\_{a-^l)  +  c){a^-l)-c){a-\-c-l)  {b  +  c-a 


w 


The  attention  of  the  student  is  again  called  to  the  fact 
that  the  line  which  separates  the  numerator  from  the  de- 
nominator of  a  fraction  is  a  vinculum  or  bracket.  Hence, 
he  will  apply  the  rules  of  brackets,  Arts.  51  and  55. 

123.  Expressions  may  occur  involving  both  addition  and 
subtraction. 

Ex.    Find  the  value  of  2  +  ^^^,'-^. 

Ans     2(a^-~^^)  +  (^'4-y)-(a-Z>)(^-^)     ^ci'-^'^al-W 


Examples — 21. 

Find  the  value  of 

a       {a—l)      a       (a  +  I?)  ^  Sa—4:b     2a—h—c    15a— 4c 
Yb'^oTb)  '  Yb'^3  (a-h) '  ~"2  3        '^~T2~"' 


a 


a  b  a  b        a—b        ah 


^'   a-b     ^'    a  +  b'^  a-b'   a-b     a  +  b'  a  +  b'^  a'-b' 


a      (ad—bc)x^  a^  +  b"^     a—b     2x'^—2xij+i/^        x 


c       c{c+dx)  '  c^—b^     a-\-b^        x^—xy  ^—y' 

1  1  ^      .  ^  — ^     C'—a    b—c 

••  2{a-x)'^2{a-\-x)  '^  a'-^-x"  '  ~W^~^^'^~bG~' 

^^ 1 1  1  _1_        1  3a 


2{x-l)     2[x-\-l)     x''  '  2a  +  b'^2a~b    ^a^-V 


92  ELEMEKTARY    ALGEBRA. 

a      {a!'  —  lf)x     a(a^  —  l'')x^    ^     \_    1 x—\ 


x^—y      x—y     x  +  y  x  +  y     x  ~y^     x  -j-y 


11     ^_     ^+^  12.  2     ^'""^'  •  ^^^' 


13.     -^ 7 r.  14.    ^ ; V-f j. 

a     a(a—x)  y        x+y     xy—y^ 

^^       X             x^  x? 

15. -r, ri-  +  - 


1-x    {1-xy  "^  (1-^)'* 

16.  ^-v+.     '  '-^ 


8(1-^)   '  8(1+^)      4(1  +  ^')' 

Bemarlc. — In  the  preceding  examples  we  have  combined  two  or  more 
fractions  in  a  single  fraction.  On  the  other  hand,  we  may  if  we  please 
break  up  a  single  fraction  into  two  or  more  fractions.    For  example : 

8Z>c— 4ac  +  5a5__35c    4<xc    ^ab 3      4      5 

abc         ~abc    abo    abc     a      b      c' 

but  the  beginner  must  not  confound  -; —  with  -r . 

b—c  be 

124.  The  addition  and  subtraction  of  fractions  can  often 
be  much  simplified  by  observing  closely  the  factors  of  the 
denominators,  and  avoiding  unnecessary  multiplications  in 
reducing  the  fractions  to  a  common  denominator. 

Ex.  Find  the  value  of 

a  l  c  ^ 

{a—d){a  —  c)     {b~c){b—a)     {c—a){c—'b)' 

Here  the  beginner  is  liable  to  take  the  product  of  the  de- 
nominators for  the  common  denominator,  and  thus  to  ren- 
der the  operations  laborious. 

How  tlie  addition  and  Bubtraction  of  fractious  may  often  bo  simplified. 


ADDITIOIs^   AKD    SUBTEACTIOiT   OF   FBACTIOXS.  ^3 

The  second  fraction  contains  tlie  factor  l—a  in  its  denom- 
inator, and  this  factor  differs  from  the  factor  a  —  l),  which 
occurs  in  the  denominator  of  the  first  fraction  only,  in  the 
sign  of  each  term ;  and  by  Art.  110 : 

1)_ I 

{b-c){b-a)~     {b-c){a-l))' 

Also,  in  the  denominator  of  the  third  fraction,  by  the 
Eule  of  Signs  we  have, 

(c  —  a){c—'b)  —  {a—  c)  {b—c). 

Hence,  the  given  expression  may  be  written, 

a  b 


+  - 


{a-b){a-c)     {b-c){a-by  {a-c){b-c)' 

And  in  this  form  we  see  at  once  that  the  L.C.M.  of  the  de- 
nominators is  {a'-b){a—c){b—c). 

By  reducing  the  fractions  to  this  lowest  common  denomi- 
nator, we  get 

a{b—c)  —  b{a—c)-\-c{a—b)     ab  —  ac'-ab  +  bc^-ac—bc_ 
{a—b){a—c){b—c)        ~~      {a—b){a—c){b  —  c) 

Examples — 22. 
Find  the  value  of 
a 


1. 


{x—a){a—b)     {x—b){b—ay 
b' 


3, 


{x-a){a-b)  ^  {x-b){b'-a) 
1  1 


{a-b){a-~c)'^  {b-a)(b-cy 


^4  ELEMEIS-TARY    ALGEBEA. 

a 


^'      {a-~b){a-c)  "^  {h-a)(})-G) ' 


6.       .       .^.         .-f-  ^ 


7. 


a{a—b){a—c)       h{b—a){h—c)      abc 
{a-b){a-c)  "^  {b-a)  {b-c)  "^  (c-a)  (c-b)' 


XYII.    Multiplication  of  Feactions. 

125.  To  multiply  fractions  together; 

Multiply  the  numerators  together  for  a  neiu  numerator^ 
and  the  denominators  for  a  neiu  denominator. 

Suppose  that  we  have  to  multiply  -r  hy  -r :   let  —  =0?, 

/> 

-j—y\    :.   a—bx,   c—dy,   and  ac=bdxy\     hence  (dividing 
a 

ac  a      c 

each  of  these  equals  by  bd),  j-^^xy,  but  xy—~y,—^  and 

ac    aXc      product  of  numerators       ,  ,.     ,     ,, 

T-i=Ti — i^-^^—j — T-i-^ — 7 — y  whence  the  truth  of  the 

bd    bXd    product  of  denominators 

rule  is  manifest. 

Similarly  we  may  proceed  for  any  number  of  fractions. 

a^b    a-b    3_3(a+^)(o^~^)     dja'-V) 
^^'      c  +  d^c-d^2~2{c-\-d){c-d)'~'2{c'-d')' 

126.  The  Kule  of  Signs  (Art.  61)  gives  the  following  re- 
sults in  the  multiplication  of  fractions : 

To  multiply  fractions  together. 


MULTIPLICATIOI!^   OF   FRACTIO^^S.  ^ft 


a  c  _a      —  c     —  ac_     ac 


a      c      ~  a     c      —  ac         ac 


h^  d~  b    ^  d~  bd  ~     bd' 


a  c      —  a     —  c_ac 

b  d~  b         d  ~~bd' 

127.  We  shall  now  give  some  examples.  Before  multiply 
jng  the  factors  of  the  new  numerator  together,  and  the 
factors  of  the  new  denominator  together,  examine  if  any 
factor  occurs  in  both  the  numerator  and  denominator;  in 
which  case  it  may  be  struclc  out  of  both,  and  the  result  will 
be  more  simple.  Art.  116.  (See  method  of  cancelling  in 
Arithmetic — Vulgar  Fractions.) 

Ex.  1.    Multiply  a  by  — . 
c 

a     a      b     ah 
Hence  a—  and  —  are  equivalent;  so,  for  example, 

.    X  4:X  T      1   /^  0\  ^^  —  3 

4— =r-^;  and  — (2x— 3)  =  — - — . 
5      5  4^  ^         4 


Ex.  2.     Multiply  -  by  ^. 

X      X  _xXx_x^  ^ 

y    y'vy^'f' 

thus  (— )    =-a. 

^yJ    y 

Application  of  the  Rule  of  Signs.    Cancellation, 


96  eleme:n'taey  algebka. 

Ex.3.    Multiply  I?  by  |. 

Sa     Sc_3aXSc__2cXl2a_2c 
4:b     9a~4:bx9a~dbxi2'a~3b' 

Ex.  4.    Multiply  ,    ,  ,,,  by  -^—z.-^- 

(a+^)'^      3ab      ~  b  {a  +  b)Xda  {a  +  b  ~b{a-\-b)' 

^      ^      a     ex     a      c     ac 
Ex.  5.    tX-t=-tX-j=j--,* 
bx     a      b      a     bd 

^  5fl^.T     xy-^-if^ _J)a  {x-^ij) _ hax 4-  5^-?/ 

3c;z/     x'^~xy~'3c{x—y)~  3c{x~yy 

-^  4fl^a;     a^ — o;^     ^c  +  bx  _  4:r  ((7- + x)  _  4rir.T  +  4^^ 

3bt/     c^—x^     a^—ax~Sy{c—x)~oy{c—xy 

Bemark. — The  student  slioulcl  leave  the  denominators  of  fractions 
■  with  their  factors  unmultipUed^  as  in  Ex.  6  and  7,  unless  they  combine 
very  simply.     The  convenience  of  this  will  be  found  in  practice. 

Ex.  8.    Multiply  4  +  -+1  by  -^+--1. 
^  ''    b      a  '^    b      a 

b      a       ~ab    ab     ab~'        ab        ^ 


a      b     -,  _^'     ^'      ab ^a^-^-b"^  —ab 
b      a  ab     ab    ab~        ab 


^_±^±ab    a'-\-b'-ab_{a''-\-b''-\-al))  (a'  +  b'-ab) 
ab  ab       "~  a'^b^ 

__{a'  +  by-a'b'_a'  +  b'-^a''b' 
""  a'b'  -        a'b'       • 


MULTIPLICATION   OF   FRACTIONS.  97 

Or  we  may  proceed  thus: 

fa      h     ^\  f a      h     ^\      fa      hy"     ^ 

therefore, 

fa      h  ^  ^\  fa      I      ,  \     a\  ^     h^     ,     a"     h\  ^ 
\  0      a       J  \h      a       J     0  a  la 

The  two  results  agree,  for  -p,-\ — ^  +1  = 


2  7,2 


Examples— 23. 

Find  the  yalue  of  the  following : 

M     ^hc  ^    h^     c^ 

ob     oa  00    ac    ah 

a^h     Vc     c'a  .    x-{-l     x-h2       x—l 

0»        2      X      ^  X  "^     •  ^»  T  X      2         -1   X  V        ■    L-i \  i» 

i?;i/    y  z     zx  ^—1^—1     (a;4-2j 

cc  +  a     Va      xy  \       b  /  \       a/ 


'■  (»+^J  (*-^)- 


cc(6i^— cc)  a(a  +  x) 

2X- 


a^  +  2a:?;  +  ^^     ^^ — 2ax  +  2;^ 

^'— «/•  ^'  +  ^'  ^  +  y 

x'—(a-^h)x-}-ah    x^—c^ 

J.l/.        -~       '^        I       r  -  X      2         72' 

X  —{a  +  c)  x  +  ac    x—b 
5 


98  elemejs^taky   algebra. 

X  -\-y       \x—y    x-\-yJ 

13.  f«  i_4_i)xfi — ¥_). 

\oc    ac    ab     a)     \       a  +  o  +  cj 

^^    fx""    a"     X      a     .  \     fx      a\ 

13.  (-T+— +  l)x( . 

\a      X      a      X       J      \a      X  J 

^  .    f  X      a      tf      h  \      ( X      a      y      b\ 

14.  ( +  ^ )X — T  +  -)- 

\a      X      b      y  J     \a      x      by) 


x^—%x-\-\     x''—4x_j-^    x^  —  6x+9 
'  x'-6x-{-6^x'-4.x+3^x'--3x-}-2' 


XVIII.    Division  of  Feactions. 

128.  Eule  for  diyiding  one  fraction  by  another : 

Invert  the  divisor,  and  proceed  as  in  Multiplication. 
The  following  is  the  usual  demonstration  of  the  rule. 
Suppose  we  haye  to  divide  "t-  by  -7 ;  put  -j-=x,  and  -T=y'y 

then,  a = bx,  and  c=dy'y 

and,  ad=bdx,  and  bc—idy\ 


therefore. 


ad    bdx     X 
bc~~bdy~  y* 


-n   ,  X  a      c 

y  -^     b      d^ 

.,       «  a      c     ad     a      d 

therefore,  -r-f--7=  r=irX— • 

b      d      be     b      c 


To  divide  one  traction  by  another. 


diyisio:n"  of  feactioxs.  99 

129.  The  results  giyen  in  Art.  125  give  us  the  following  in 
connection  with  Division  of  Fractions  : 

o-  ^  c         ac        ^       a      c         ac 

b  d         M  Id         bd 

,  ac         c      a         ^      ac      c  a 

bd         d      b  bd      d  b 

•  T        .  a  c     ac  ,         ac  c  a 

Also,  since  — 7-  x  — 7=^-75   we  have  :r-,-. -—  —  —. 

b  d     bd  bd         d  b 

130.  The  student  should,  in  the  division  of  fractions,  en- 
deavor to  simplify  the  operations  as  much  as  possible  by 
striking  out  factors  which  occur  in  both  numerator  and 
denominator. 

Ex.  1.     Divide  a  by  — . 

__  a         a      b  _a      c  __ac 

Ex.  2.    Divide  ?f  by  I?. 


3a_^9^_3^    8£_2^ 
4.b'^SG~^b^'^a~W 

Ex.3.    Divide  ^^,  by      ^' 


{a-{-by  ^  a'-b'' 
ab-y         W        ab-b'     a' 


{a+by  '  a'-b'~{a  +  by^    b' 

_b  (a-b)  (a+b)  {a'-b_{a-by 
""  b\d^by  ~b{a'\-by 

x^^xy  ^   x'-if  jc^^-xy     ix-yy _     x 
x-y      (x-yy       x-y       x  -y'      x  +/ 


100  ELEMENTARY    ALGEBRA. 


E] 

5:amples — 24. 

Divide 

4.xyz'  ^  3xYz' 

1  1  6{ab-I?')  W 

x'-y'  ^  x-y  a{a  +  bY     ^^  a{a'-b'j* 

,     a'-4:x'  ,      a'-^ax        ^        82;'    ^  Ix" 


a^4-4a:c    "^  ax  +  4a;^*  *    ^'— ^        x'  +  xy-{-y'^' 

iz;^  +  ^^  -^  x^—xy^y^' 

o^-\-(a-k-c)x-^ac  ^     x^—a^ 
x'+{h  +  c)x^c  ^  ?^'' 

10.  ?!i:^I&:  by    ^'-^^ 


a;  4-^  x—xy-\-y* 

I.  (i.|)(.-£)b,J^.. 

13.     5x^-4  by  :r  +  -^.        14.     a'-\hj  a---. 

15.     — 5  by .        16. Sa-\ ^  by  a; . 

a       X     ''    a      x  a  x       ^  x 

x"      1  a:      1       1        _     a;\  ^     ^r'         x     ^      a 

17.     — , by-jH — .      18.     -^+l  +  -,by 1+— . 

y      X     ^  y^      y      X  a'^   ^ x^    ^    a       ^  x 


COMPLEX    i^RACTIOXS.  101 


XIX.    Complex  Fractions  and  other  Eesults. 

131.  Hitherto  we  have  supposed,  in  the  chapters  on  Frac- 
tions, that  the  letters  represented  whole  numbers,  but  when 
we  come  to  interpret  the  multiplication  of  fractions  we  must 
extend  the  meaning  of  the  term,  as  we  have  done  in  Arith- 
metic.    Thus,  to  multiply  "t-  by  -7,  the  fraction  -j-  is  divided 

into  d  equal  parts,  and  c  such  parts  are  taken.     Now  if  -7- 

be  divided  into  d  equal  parts,  each  of  these  parts  is  y-^;  and 

etc 
if  c  such  parts  be  taken,  the  result  is  j-y   Then,  too,  to  divide 

a  c 

■j-^J  -J  "ineans  to  find  a  quantity  such,  that  if  it  be  multi- 

C  Cb 

plied  by  -^  the  product  shall  be  — . 

132.  N^ow  with  our  extended  definitions  we  can  easily 
prove  that  all  the  rules  and  formulas  given  are  true  when 
the  letters  denote  any  numbers  ivliole  or  fractional.     Take, 

for  example,  the  formula   — =— ,  and   suppose  we  wish  to 

show  that  this  is  true  when 

a——.  0—-,  and  c=-, 
71         q  s 

^   ^  a  _m  ^  p  __m     q  _mq 

on      q     n     p     np 

Also  ac=^  — ,  and  hc~  —, 

ns  qs 

rpi  ac _mr    pr _vir     qs    mrqs    wq 

he ~ ns  '  qs  ~ ns     pr ~ nsjjr  ~ np ' 
Thus  the  formula  is  proved  to  be  true. 


-1 

2-x 

2 

2- 

2-x 

4a;  - 

~  Ax  ' 

~    2 

~  8x  " 

102  ELEMEKTAKY    ALGEBRA. 

133.  Complex  fractional  expressions  may  be  simplified  by 
tbe  aid  of  rules  respecting  fractions  which  have  now  been 
given. 


Ex.  1. 


Hence  obserye  that,  when  a  complex  fraction  is  pnt  into 

the  form  of  a  7; — -r — ,  the  simple  expression  for  it  will  be 
traction  ^         ^ 

found  by  taking  the  product  of  the  upper  and  lower  quanti- 
ties, or  extremes,  for  the  numerator,  and  that  of  the  two  mid- 
dle ones,  or  means,  for  the  denominator ;  and  that  any  factor 
may  be  struck  out  from  one  of  the  extremes,  if  it  be  struck 
out  also  from  one  of  the  means. 


Ex.  2.                                        Ex.  3. 

2x                                       20-ri; 

2x 

1            6x              6-ix        4          60 -3a: 

^-3 

~3aj-l~"3^-r          x  +  l^~~dx+4.~4:{3x  +  4:y 
3                                              3 

Ex.  4. 

a+h    a-h     {a  +  hy+ia-hy    2a'+2h' 

a-\-b    a-b     {a+by-{a-by        Aab           2ab  ' 

a-b    a  +  b            a'-b'              a'-b^ 

Ex.  5. 

n    1 . 

0—a 

Simplification  of  Complex  Fractions. 


COMPLEX   FRACTIOJ^S. 


1U3 


3  —  a~~d-~a    3—a~      d—a 


3-a' 


3  — a    4a    3— a    3-\-3a 

a  +  --r-  ^-r-] T— = -. . 


,  3  +  3a    1         4     _     4 
•      4     ■~1^3+3a~3+3a' 


Ex.  6.    Find  the  value  of  ^ when  x= 


Here  a—x=a~ 
and        l—x—1)- 


ab       a^  +  ah—ah        a^ 


ab 
"aVb 


a-\-b         a+b' 

b^ 

=  ab\-V—ab=^ r. 

a-^b 


Hence, 


a—x     a-hb  _  a^ 
l^"    b'    ~J'' 


a  +  b 


Simplify 

1 

1. 


l-\-x 

1   1  ' 

1  +  X 

1 

X 


Examples — 25. 


':+.=" 


2. 


l  +  iz;     1— a; 


1— ^     l  +  a; 


3x    x—1 


5. 


2"*""  3 


-(.H-l)---2i 


1-. 


3. 


1+- 


a;-l  + 


x—6 


X-2  + 


x-6 


104  ELEMENTARY    ALGEBRA. 

Find  the  yalue  of 

-,     x—a      x—b     ,  a^ 

7.    — ^ when  x= ~. 

o  a  a—h 

TT  when  x=--}- -^. 

a-\-b  b{b  +  a) 


8.     -+T^ 
a     o—a 

9, 


a'x-hh'ij     .  2,^2 

■ — ~  when  a—  —  and  h=  — . 

i?:  +  .^  3  3 

10.    -— — — ^ ^— „  when  t/  =  — . 

ic+y     x—y     X  —y  ^      4 

^^      x  +  2a     x—2a        4:ab         .  ^5 

J-J-.     ^77 +?rr"^ TU "2  when  x= -. 

2b  — X    2b+x      ^W—x^  a  +  b 

^^      fx—aV     x—2a  +  b     ,  a  +  b 

l/c.     I — -1 ; —  when  0:=— -— . 

\x—bj       x  +  a—2b  2 

x+y—1  a-hl  ,        ab  +  a 

lo.     ■ — — -—  when  xz=   ,     ^  ,  and  y=—, — -. 

x-y  +  1  ab-\-V  ^     ab-hl 

134.  The  following  results  should  be  noticed. 

If  —==—-.  then 
b      d 

^  ,  ci  c  b      d  ,..     abbe  a      b    ,... 

^--b=^-d'  ^^  ¥=7«'  7X7=7X7'  ''  7=7  (">' 

a  c  a  +  b    c-\-d  ,.... 

CI'     ^      0      ^        a—b    c—d  ,.  . 

a:^b      b      c^d     d         a:^b     r 
hence  -7-X— ^-^-X— ,  or = 

/;  a  an  a 


iXYOLUTio:Nr.  105 

,  a-^l        h         c-\-d        d  a  +  b     c{-d,  .^ 

and         — T— X -j= — j-X 7'  ^^^ 7= 7(^1): 

0        a—o        d       c—d         a—o     c—d^    ^ 

and  any  of  tliese  last  may  be  inverted  by  (i),  or  alternated 

,      ,...     ,^  ct  c        a     a^h     a  +  b     a—h     . 

by  (n);  tlms,  -^^=y^^j,  -=^-^,  ^=^=5'  &'^- 

So  that,  If  any  tivo  fractions  are  equal,  ive  may  comhin^ 
hy  addition  or  subtraction,  in  any  ivay,  the  numerator  and 
denominator  of  the  one,  provided  that  ive  do  the  same  with  the 
other. 


XX.    Involution. 

135.  The  jorocess  of  obtaining  the  poivers  of  quantities  is 
called  Involution,  A  poiver  has  been  defined  to  be  the p)roduct 
of  two  or  more  equal  factors.  All  cases  of  Involution,  then, 
are  merely  examples  of  multiplication,  where  all  the  factors 
are  the  same;  and  the  rules  given  in  the  present  chapter 
follow  immediately  from  the  laws  of  Multiplication. 

136.  Any  even  p)Oiver  of  a  negative  quantity  is  positive. 
Any  odd  power  of  a  negative  quantity  is  negative. 

This  is  a  simple  consequence  of  the  Eule  of  Signs. 

Thus,  —■aX—a—-\-a^',  —aX—aX—a=-{-a''X-'a——a^'y 
—  aX—aX  —aX  —a=—a''X  —a  —  -}-a*;  and  so  on. 

Let  the  student  notice  :  1.  That  any  eve^i  power  of  a 
quantity  is  the  same  whether  that  quantity  be  negative  or 
positive.  Thus  (  +  «)^  and  {  —  ay  are  each=+^^;  and 
{  —  {a  +  b)y  and  {  +  {a-\-b)y  are  each=  +  (a  +  ^)\  2.  ISTo  even 
power  of  any  quantity  can  be  negative,  3.  Any  odd  power 
of  a  quantity  will  have  the  same  sign  as  the  quantity  itself. 

137.  The  expo7ient  of  any  poiver  of  a  power  is  equal  to  the 
product  of  the  exponents  of  the  tivo  poivers. 

Involution— Power— Signs  of  Powers. 

5* 


106  ELEME^^TAEY    ALGEBRA. 

Thus,  tlie  cube  of  a%  that  is,  {ay=za'i  foY,{ay  =  a'Xa'Xa* 
Similarly,  {ay  =  a'';    {-ay=-a'';  {-ay=z-a'',  {a^Y 

138.  Eule  for  obtaining  any  power  of  a  monomial  ex- 
pression : 

Multiply  the  exponent  of  every  factor  in  the  expression  hy 
the  index  of  the  required  power,  and  give  the  proper  sign  to  the 
result. 

Thus,  for  example, 

{a''l)y=a'h']  {-aWy=-a'b';  {abVy=a'bV'; 

{-a'bYy=-a''b'V'',  {2ab'cy=2Vb'Y'^64:a'b''c'\ 
It  is  usual  to  raise  the  numerical  coefficient  at  once  to  the 
required  power,  instead  of  first  writing  it  with  an  exponent. 
Thus,  ( - 2xy'z') '  =  -  8xYz\ 

139.  Eule  for  obtaining  any  power  of  a  fraction :  Eaise 
both  the  numerator  and  deiiominator  to  that  potver,  and  give 
the  proper  sign  to  the  result.    This  follows  from  Art.  122. 

For  example, 

140.  Some  examples  of  Involution  of  binomial  expressions 
have  already  been  given.    Thus, 

{a-\-by=a''  +  2ab  +  b\ 

(a-'by=a'-2ab+b\ 

By  (137)  we  may  shorten  the  operation,  finding  the  4th 
power  of  a  quantity  by  squaring  its  square ;  and  similarly, 
to  find  the  6th,  8th,  &c.  powers,  we  may  square  the  3d,  4th, 
&c.  powers. 

Rule  for  obtaining   any  power  of  a  monomial  exprebsion ; — of  a  fraction.    In- 
volution of  binomial  expressions. 


IXYOLUTIOX.  107 

So  also  to  find  the  cube  or  3d  power,  vre  may  take  the 
product  of  the  quantity  itself  and  its  square ;  to  find  the  5th, 
we  may  take  that  of  the  square  and  cube,  &c. 

Thus  we  shall  have, 

(a-by={a'-2ab  +  b'){a-b)=a'-3a'b  +  3ab'-b'; 
{a+by  =  {a'  +  2ab+b'){a'  +  2ab  +  ¥)=a'  +  ^a'b+ea'b* 

+  4.ab'  +  b'; 
{a-by={a'-2ab  +  b'){a''-2ab  +  b')=za*-'4.a'b  +  6aV 

^4.ab'  +  b'; 
{a+by={a  +  by{a-\-by=a'  +  6a*b  +  10a'b'-hl0a'b'-^6ab^ 

(a-by={a-by{a-by=a'-6a*b+10a'b'-10a'b'-\-5ab* 
-b\ 

The  student  should  remember  the  above  results,  though 
the  higher  powers  of  binomial  expressions  are  best  obtained 
by  the  Binomial  Theorem,  which  we  shall  give  subsequently. 

It  will  be  noticed  in  the  above  examples  that  any  power 
oi  a—b  can  be  immediately  obtained  from  the  same  power 
of  a-\-b  by  changing  the  signs  of  the  terms  which  involve 
the  odd  powers  of  b. 

141.  The  results  of  Art.  140,  can  readily  be  applied  to  tri- 
nomial  expressions, 

Ex.  1.     {a  +  b^cy=a'-Y2a{b^-c)^{b+cy 

=a''  +  2ab  +  2ac-{-b'-\-2bc-{-c\ 

Involution  of  trinomial  expreseions. 


108  ELEMEKTARY    ALGEBEA, 

Ex.  2.     {a-\-l)  +  cy={a-\-{'b^-c)Y 

Ex.  3.     (a-^-c)^={^-(Z»  +  c)}^ 

==a'-3^'^-3aV  +  3a5'  +  6rtJc  +  3r^c'~Z»'-3/;'c 
-3Z^c^-6'^ 
Or  thus : 
(a-l-cy={{a-l)-cY^{a-l>y-^{a-'byc-{-^a-l)c^-c\ 

which,  of  course,  when  expanded,  would  give  the  same  re- 
sult as  before. 

Ex.  4.    {^x-dy={2xy-4. .  3 .  {2xy  +  6 .  31  {2xy-^ .  3^(2a:)  +  3* 

Examples — 26. 

1.  Eind  the  yahies  oi  {2aiy,  (-3a^^V)',    (-^T. 

Write  down  the  expansions  of 

2.  {x^%)\         3.   {x-'^y,        4.  (^+3)^        5.  (l  +  2a;)*. 
6.    (2m-l)'.      7.  {3.r+l)\      8.   {9.x-ay.      9.  (3.'?;  +  2a)*. 

10.  (4a-3Z>)'.  11.   {ax-yy.  12.  (ax  +  rr^)'. 

13,  {%am-my,  14.  (a-Z>  +  c)«.  15.  (l-a:-f .t'*)'. 


IXYOLUTIOX.  109 

142.  The  square  of  imy  2)oIl/nG}mal  expression  may  be  ob- 
tained by  either  of  two  rules.     Take  for  example, 

{a  +  b  +  c+dy. 
We  will  find, 

{a  +  b  +  c-hcl}'' 

= a'  +  Z>'  +  c'  +  cr  +  2al)  +  2ac  +  2ad +2bc  +  2bd+ 2cd. 

We  see  from  this — the  square  of  any  polynomial  may  be 
found  by  setting  down  the  square  of  each  term.,  and  then  the 
doiihle  2^rodiict3  of  all  the  terms,  talcen  two  and  two, 

iVgain,  we  may  put  the  result  in  this  form, 

{a^l^c  +  dy 

^a''-\-2a{l)  +  c  +  d)-Vl)'^2'b{c-Vd)+c'^2ci:  vd^. 

and  this  may  be  obtained  by  the  following  rule : 

The  square  of  any  multinomial  expression  consist i  df  the 
square  of  each  term,,  together  ivith  twice  the  product  \  f  cadh 
term  by  the  sitm  of  all  the  terms  ivhich  folloiv  it. 

Ex.  1.    {l  +  2x-\-^xy^l-\-2{2x  +  'dx')+^x^-\-ix{^x'')-^.&J 

=  l-\-4cX-\-10x'-\-l2x'  +  ^x\ 

Ex.  2.    {l-2xy=[{l-2xyY^{l'-^,x-\-12x''-^x'Y 

=:l-12x-\-2ix''-Ux^ 

+  144^^-1920;*  f  64^" 
=  l-12.^'  +  602;^-1602;^4-240.^'^-192a;*  V^^x\ 


The  square  of  polynomial  expreseiona,— two  rules. 


110  elementary  algebra. 

Examples — 27. 

Find 

1.  {a  +  h-^c-\-dy-{a-'b  +  c-d)\ 

2.  {a  +  d-^c  +  dy+{a-b-\-c-dy. 

3.  {1  +  x-hxy.  4.     {l--x+xy.  5.     (l+2;-ic^)\ 
6.     {l-\-3x-h2xy.                           7.     {l-3x+dxy. 

8.  (2  +  3a;+4a;^)^+(2-3a;  +  4:c^)^ 

9.  (l_:c+a;^+a;y.  10.     {l  +  2x+3x'+^xy. 

XXI.    Eyolution. 

143.  Evolution  is  the  inverse  of  Involution.  Evolution  is, 
then,  the  method  of  finding  the  roots  of  quantities.  It  is  usual 
in  this  connection  to  use  the  word  extract  in  the  same  sense  as 
find.  Thus  to  extract  the  square  root  is  to  find  the  square 
root, 

144.  It  follows  from  (136)  that— 

1.  Any  even  root  of  a  positive  quantity  will  have  the  douUe 
sign  db. 

Thus  the  square  root  of  a^  is  ±a,  the  fourth  root  of  a*  is 

2.  Any  odd  root  of  a  quantity  has  the  same  sign  as  the 
quantity  itself 

Thus,  for  example,  the  cube  root  of  a^  is  a,  and  the  cube 
root  of  —a^  \^  —a, 

3.  There  can  he  no  even  root  of  a  negative  quantity. 
Hence  the  indicated  even  root  of  a  negative  quantity  is 

called  an  impossible  quantity  or  imaginary  quantity,     ^ —a^, 
^—a,    "^  —  1,  are  imaginary  quantities. 

Evolutiou.    Three  Rules  for  the  Signs  of  Roots.    Imaginary  quantities. 


EVOLUTION-.  Ill 

145.  Eule  for  finding  any  root  of  a  monomial  integral 
expression.  Extract  the  required  root  of  the  numerical  coef- 
ficient, divide  the  expo7ient  of  each  literal  factor  hy  the  index 
of  the  root,  and  give  the  proper  sign  to  the  result. 

Since  the  cube  power  of  a^  is  a^,  therefore  the  cube  root  of 
a^  is  a^,  and  so  on. 

Thus,  for  example,  V {Ua^h')=  V {^^a''h')  =  :^4.a'b\ 

V{-Sa'b'c'')  =  V{-2'a'bY')  =  -2aWc\ 

V{266xy)  =V{4:'xy)  =  ±:4.xy\ 

146.  To  obtain  any  root  of  a  fraction:  Find  the  root  of 
the  numerator  and  denominator,  and  give  the  proper  sign  to 
the  result. 


For  example,  \/ {^)=\^  {-^) 


2a 


147.  Suppose  we  require  the  cube  root  of  a^.  In  this  case  the 
exponent  2  of  the  quantity  is  not  divisible  by  the  index  3  of 
the  root ;  then  we  cannot  find  the  root  of  it,  but  can  only 
indicate  that  the  root  is  to  he  extracted  by  writing  it 
thus,  V^.  Similarly,  \f~^,  v/^,  V^,  indicate  roots  which 
we  cannot  extract.  Such  quantities  are  called  surds,  or 
irrational  quantities;  the  difference  between  surds  and 
imaginary  quantities  being  that  surds  have  real  values, 
though  we  cannot  find  them  exactly,  while  there  cannot  be 
a  quantity,  positive  or  negative,  an  even  power  of  which 
would  produce  a  negative  quantity. 

Examples — 28. 
1.     Find  the  square  roots  of  ^a^h'c\  4t9xyz%  100a'b'Y\ 

Eule  for  finding  any  root  of  a  monomial  integral  expression;  of  a  fraction. 
Surds,  or  Irrational    quantities. 


115>  ELEMENTARY    ALGEBIIA. 


_      ^.    ,  .,  .      .  ^a'x\f    4.9xy    mxY' 

2.     Find  the  square  roots  of    ---4-  ,  -777-^-  ,  -ttt-ott- 


27^;^''   ^125a^'^   ^  343      * 


3.     Find  v/-^^4^,  V-^,,  V^,T.,  V 


*■  -  v(w>  i/(5^.>  ♦/(^:-> 

148.  To  find  the  square  root  of  a  polynomial:  We  hnnw 
that  the  square  of  a  +  ^  is  a^-\-2ah-\-lf.  Let  us  observe,  then, 
how  from  a'^  +  2ab-hh^  we  may  deduce  its  square  root  a  +  b. 
This  will  lead  us  to  a  general  method  of  finding  the  square 
root  of  polynomial  expressions. 

a'+2ah  +  I)'{a+b 
a^ 

2a  +  b)2ab  +  b' 
2ab+b'' 


Arrange  the  terms  according  to  the  powers  of  one  letter,  a; 
then  the  first  term  is  a^,  and  its  square  root  is  a.  Subtract 
the  square  of  a,  that  is,  a^,  from  the  whole  expression,  and 
bring  down  the  remainder  2ab  +  b^.  Diyide  2ab  by  2a  and 
the  quotient  is  b,  which  is  the  other  term  of  the  root;  lastly, 
if  we  add  this  b  to  the  2a,  multiply  the  2a+b  thus  formed  by 
b,  and  subtract  the  product  from  2ab  +  b'^,  there  is  no  re- 
mainder. 

Now  we  may  follow  this  plan  in  any  other  case,  and,  if  we 
find  no  remainder,  we  may  conclude  that  the  root  is  exactly 
obtained. 

Ex.1.  Ex.2. 

9x'  +  exy+y\3x  +  y  16a'-56ab  +  4:9b'{4.a-'7b 

i)x+y)  Qxy-Vy''  8a-7Z>)-5G^/Z»  +  49Z'' 

^xy-^y''  — 56«Z>  +  49Z>' 


To  find  the  square  root  of  a  polynomial. 


evolijtio:n'.  113 

Ex.  3.        ^a^-^ah-F^^a-l) 

^4.ab  +  b' 
-2b\ 
Here  we  find  a  remainder  —25"^ ;  we  conclude,  therefore,  that  2a— b 
is  not  tlie  exact  root  of  Aa^ —Aab—b'^  which  is  a  surd,  and  can  only  be 
written  V^a'_4^ab-b-'' 

149.  If  the  root  consist  of  more  than  two  terms,  a  similar 
process  will  enable  iis  to  fir 4  it,  as  in  the  following  example, 
where  it  will  be  seen  that  the  divisor  at  any  step  is  obtained 
by  doubliiig  the  quantity  already  found  in  the  root,  or 
(which  amounts  to  the  same  thing  and  is  more  convenient 
in  practice)  by  douhling  the  last  term  of  the  ])receding  divisor, 
and  then  annexing  the  neiu  term  of  the  root, 

Ex.  162,-'-24^'+252;*-20a;'-fl0a;'-4::c+l(42.-'-32;^  +  2a;-l 

—24:x'-{-   Qx' 

8x'-6x''  +  2x)  16x'-20x'-\-10x^ 
16x'-12x^ji^_i^ 

8x'-Gx''  +  4:X-l)-  Sx'+   i5x''-4:X  +  l 
-  8x'-\-   6x''-4:X-\-l 

150.  It  has  already  been  remarked  that  all  even  roots  have 
double  signs.  Thus,  the  square  root  of  a'^-\-2ab  +  b^  may  be 
—  {a+b),  that  is,  —a—b,  as  well  as  a-\-b;  and,  in  fact,  the 
first  term  in  the  root,  which  we  found  by  taking  the  square 
root  of  tt^  might  have  been  —a  as  well  as  a,  and  b}^  using 
this  we  should  have  obtained,  also,  ~~b. 

So  in  (148)  Ex.  1,  the  root  may  also  be  —3x—y;  in  Ex.  in 
(149),  —4:X^  +  dx'^~2x-{-l;  and  in  all  such  cases,  we  should 
get  the  two  roots  by  giving  a  double  sign  to  the  first  term  in 
the  root. 

When  the  root    consists  of  more  than  two  terms.    Double  Signs. 


114  ELEMENTARY    ALGEBRA. 

151.  As  the  4tli  poiver  of  a  quantity  is  the  square  of  its 
square,  so  the  4tli  root  of  a  quantity  is  the  square  root  of  its 
square  root,  and  may  therefore  be  found  by  the  preceding 
j*ule.  Similarly,  the  8th  root  may  be  found  by  extracting 
the  square  root  of  the  4th  root. 

Thus,  if  it  be  required  to  find  the  4th  root  of 

a'  +  4a'ic+ 6«^V  +  4«a;'  +  x\ 

the  square  root  will  be  found  to  be  a^-\-^ax-\-x^,  and  the 
square  root  of  this  to  be  a+ic,  which  is  therefore  the  4th  root 
of  the  given  quantity. 


Examples — 29. 

Extract  the  squnre  root  of 

1.  a;'+2a;'  +  3a;'  +  2a;  +  l.  2.     1— 2^  +  5a;''-4a;''  +  4ct . 

3.  x'-V^x^  +  '^bx^^-^^x+U.      4.    x'^4.x'-V%x-\-4:. 

5.  l-^x^l()x^-  12x^ + 9a;*. 

6.  4x'--4x''-7i«*  +  4x'  +  4. 

7.  x'-2ax'-\-6aV-4.a'x+4.a\ 

8.  x'-2ax'  +  {a'  +  2I}^)x''-'2ab'x  +  b\ 

9.  ic*-12a;'+60^*-160:z:'  +  240:r'-192a;+64. 

10.  x'  +  ^ax'-10aV-\-4:a'x  +  a\ 

11.  l-2x-{-dx'-4.x'  +  6x'--4:x'  +  3x'-2x''  +  x\ 

^x"     X      l^x^      9f_     6xy    16^ 
9p~7~l5y^^l6?"^  57"^25;2^' 

The  fourth  root,  etc. 


EVOLUTIO:^'.  115 

Extract  the  4tli  root 

13.  Of  l-4:X-{-6x''-4.x'  +  x*  and  of  a'-8a'-f  24a'-32«4-16. 

14.  0£16a'-96a'b  +  2Wa'b'-216ah'-i-SU\ 
Find  the  8th  root 

15.  Of  {x'-2x'y-{-3xY-2xy'-{-t/}\ 

152.  The  observation  of  the  square  roots  of  trhiomial  ex- 
pressions  enables  us  to  find  the  square  root  of  complete  (/.  e.), 
exact  squares  of  these  terms  very  easily,  without  going 
through  the  entire  process  of  Art.  148. 

EuLE. — Arrange  the  terms  according  to  the  powers  of  some 
one  letter.  Find  separately  the  square  roots  of  the  extreme 
terms,  and  take  their  sum  or  difference  accordingly  as  the  sig?t 
of  the  middle  term  is  -h  or  — . 

Thus,  a^  +  2ax  -f  2;*  is  a  complete  square  arranged  according 
to  powers  of  a,  and  its  square  root  is  ^a^~^  ^x^,  or  a-\-x, 
.*.  a-{-x  squared  produces  a^  +  2ax  +  x^.  The  square  root  of 
a^—2ax  +  x^  is  a—x,  for  the  same  reason. 

Ex.  1.  ^¥~+l-{-2a='^a'  +  2a-hl=^^a^-{-^l=a-hl. 
Ex.  2.  ^¥~-{-9-6x=  y/x''-6x  +  9=  v^?-  >/d=x—S. 
Ex.  3.        v^4Ty^-4^=  s/y^-4:y  +  A=:  n/^-  \^I=y-2. 

Ex.  4.       \/x^-px+^=  v/i^_f^^^_|. 


Ex.  5.       \/x'  +  3x+\=^x'^]/\=x-{-%. 

To  find  the  square  root  of  complete  trinomial  squares. 


116  ELEME^'TARY     ALGEBRA. 

Ex.  6.       ^  ni'jf  +  '^iniix  +  n^  —  "^ i)fx'  +  ^ if  ~ mx  +  n. 
Ex.  7.       v'9^'7^to7/  +  a~'==  ^9xy-  ^7'r=dxy-a, 
Ex.  8.       v/i,rZ^^4-^^^;-|-c-^  =:  \/I^^"  +  V?  =  J-«Z^  4-  c. 
Find  the  square  roots  of  the  following  expressions : 
Ex.  9.       16a^  +  40aZ>+25^^  10.    49^*-84a'^^  +  36^>^ 

Ex.  11.     36:^;°  +  12i^'  +  l.  12.     64a'  +  48a/;c  +  9Z^V. 

-g      .o      25^^^^«^4^^  9x^-242^^  +  16 

^"^^       *     2ba'  +  20ac-\-^o^  '     S^~12:z:  +   9* 

153.  By  observing  the  terms  of  a  complete  trinomial 
square  arranged  according  to  one  letter,  we  see  that  the  mid- 
dle term  is  twice  the  product  of  the  square  roots  of  the  two 
extreme  terms.  Hence,  the  quantity  which  must  be  added 
to  an  expression  of  the  form,  x^  +  22jx,  in  order  to  form  a 
complete  trinomial  square,  is  the  square  of  one-half  of  the 
CO  factor,  or  coef[icient,  2p  of  x\  that  is,  ]f.  Observe  that  x 
represents  the  square  root  of  the  first  term. 

Thus,  m^x^  +  ^mnx  requires  the  square  of  the  half  of  2?^, 
or  7f^  to  complete  it,  giving  m^x^  +  2in7ix  +  ^^^ 

dx^y^—Qaxy  requires  the  square  of  the  half  of  2a,  giving 
9x'^y^  —  6axy  +  a^. 

Complete  the  squares  in  each  of  the  following  cases : 


(1.)     x'-12x  +  - 

(3.)     x'+llx+- 
(5.)     x'-  ix-h- 

(7.)  ■  x'-i-  px-h- 

7t 
(9.)     --fo+- 


(2.)     x=-|+- 
(4.)     x''—  X  +- 


(G.)     36:^;H24:?;-|— 

(8.)     16:?;'-56a;  +  — 

(10.)     4«V  +  4«^^+- 


To  complete  the  square  of  expressions  of  the  form  of  a;2  +  2/xc. 


EVOLUTIOIS^  117 

154.  The  method  of  finding  the  square  root  of  numbers 
is  derived  from  the  methods  of  Arts.  148  and  149.  (See 
V^enable's  Arithmetic — Square  Eoot.) 

The  square  root  of  100  is  10 ;  the  square  root  of  10000  is 
100  ;  the  square  root  of  1000000  is  1000,  and  so  on.  Hence, 
it  follows  that  the  square  root  of  any  number  between  1  and 
100,  lies  between  1  and  10,  that  is,  the  square  root  of  any 
number  haying  one  or  hvo  figures  is  a  number  of  one  figure ; 
ISO,  also,  the  square  root  of  any  number  between  100  and 
1000,  that  is,  having  three  or  four  figures,  lies  between  10 
and  100,  that  is,  is  a  number  of  hvo  figures,  and  so  on. 

Hence,  if  we  set  a  dot  over  every  other  figure  of  any  given 
square  number,  leginning  ivitli  the  units  figure,  the  number 
of  dots  will  exactly  indicate  the  number  of  figures  in  its 
square  root.  Thus,  for  example,  the  square  roots  of  256  and 
4096  consist  of  two  figures  each,  and  the  square  roots  of  16384 
and  6il524,  of  three  figures  each. 

155.  Find  the   square  root   of  3249.  9^00 

Set  the  dots  accordinsr  to  the  rule.     The     ^^^     ZZ^TT^ 

.  ,  .  ,     .  X        .  T   ^     100  +  7  749 

root  must  consist  oi   two  figures.     Let  ^aq 

a-^-h  denote  the  root,  where  a  is  the  value  

of  the  figure  in  the  tens  place,  and  h  of  the  figure  in  the 
units  place.  Then  a  must  be  the  greatest  multiple  of  ten, 
whose  square  is  less  than  3200,  that  is,  a  must  be  the  square 
root  of  the  greatest  exact  square  contained  in  3200.  Now, 
as  25  is  the  greatest  square  in  32,  2500  must  be  the  greatest 
in  3200 ;  hence,  a  is  50.  Subtract  a^ — that  is,  the  square  of 
50 — from  the  given  number,  and  the  remainder  is  749.  Di- 
vide the  remainder  by  2a — that  is,  by  K'O — and  the  quotient 
is  7,  wiiicli  is  the  value  of  h.  Then  {2a-{-b)b — that  is, 
(100  +  7)7,  or  107x7  =  749—18  the  number  to  be  subtracted ; 
and  as  there  is  no  remainder,  we  conclude  that  50  +  7,  or  57, 
is  the  required  square  root.  If  the  number  be  such  that  its 
root  consists  of  three  places  of  figures,  let  a  represent  the 


118  ELEME:^?TARY    ALGEBRA. 

value  of  the  liuDdreds  figure,  and  h  of  the  tens  fignie;  then 
having  obtained  a  and  h  as  before,  let  the  hundreds  and  tena 
together  be  a  new  yalne  of  a,  and  then  as  before  find  a  ne«»' 
yalue  of  h  for  the  units. 

Example.  186624  (400  +  30  +  2 

160000 

800  +  30      =830)     26624 

24900 

800  +  60  +  2=862)       1724 

1724 

Here  the  number  of  dots  is  three,  and  therefore  the  num- 
ber of  figures  in  the  root  will  be  three.  ISTow  the  greatest 
square-number  contained  in  18,  the  first  period  (as  it  is 
called),  is  16,  and  the  number  evidently  lies  between  160000 
and  250000,  that  is,  between  the  squares  of  400  and  500. 
We  take  therefore  400  for  the  first  term  in  the  root, 
and  proceeding  just  as  before,  we  obtain  the  whole  root, 
400+30  +  2=432. 

186624(432 

16  The  ciphers  are  usually  omitted  in  practice,  and  it 

83)366  will  be  seen  that  we  need  only,  at  any  step,  take  down 

__  the  next  period,  instead  of  the  whole  remainder. 

862)1724 
1724 

156.  Kule  for  finding  the  square  root  of  any  given  number : 
Set  a  dot  over  every  other  figure,  beginning         Ex.  1. 
with  that  in  the  units'  place,  and  thus  divide  3249(57 

the  whole  number  into  periods.     Find  the  25 

greatest  number  whose  square  is  contained  in      i07)  749 
the  first  period  ;  this  is  the  first  figure  in  the  749 

root;  subtract  its  square  from  the  first  period, 
and  to  the  remainder  bring  down  the  next  period.    Divide  this 
quantity,  omitting  the  last  figure,  by  twice  the  part  of  the  root 

Rule  for  finding  the  square  root  of  any  given  number. 


EVOLUTION.  119 

already  found,  and.  annex  the  residt  to  the  root  and  also  to  the 
divisor ;  then  midtiply  the  divisor  as  it  noio  stands  by  the 
part  of  the  root  last  obtained  for  the  subtrahend.  If  there  be 
more  periods  to  be  brought  doivn,  the  operatio7i  must  be  re- 
peated. 

Ex.  2.  In  Ex.  2,  notice  (i)  that  the  second  remainder,  49, 

77841  ( 279  i^  greater  than  the  divisor  47  ;  this  may  sometimes 

4  happen,  but  no  difficulty  can  arise  from  it,  as  it 

47)378  would  be  found  that,  if  instead  of  7  we  took  8 

^^^  for  the  second  figure,  the  subtrahend  would  be 

549)4941  384,  which  is  too  large.      And  (ii)  that  the  last 

figure,  7,  of  the  first  divisor,  being  doubled  in  order 

Ex.  3.  to  make  the  second  divisor,  and  thus  becoming  14, 

10291264  (3208  ^^^^^es  1  to  be  added  to   the  preceding  figure,  4, 

9  which  now  becomes  5.    In  fact,  the  first  divisor 

62)129  is  400-i-70,  which,  when  its  second  term  is  doubled, 

124  becomes  400-f  140,  or  540. 

6408)  51264  In  Ex.  3,  we   have  an  instance  of  a  cipher  oc- 

^^^^^  curring  in  the  root. 

157.  If  the  root  have  any  number  of  decimal  places,  it  is 
plain  (by  the  rule  for  the  multiplication  of  decimals)  that 
the  square  will  have  twice  as  many,  and  therefore  the  number 
of  decimal  places  in  the  root  will  be  half  that  number. 
Hence,  if  the  given  square  number  be  a  decimal,  and  one  of 
an  even  number  of  places,  we  set,  as  before,  the  dot  over  the 
units'  figure,  and  then  over  every  other  figure  on  both  sides  of 
it.  The  number  of  dots  on  the  Uft  of  the  decimal  point  tvill 
indicate  the  number  of  integers  in  the  root,  and  the  number  of 
dots  to  the  right,  the  number  of  decimal  places  in  the  root. 

For  example : 

The  square  root  of  32.49,  is  one-tenth  of  the  square 
root  of  100x32.49;  that  is,  of  3249.  So,  also,  the  square 
root  of  '003249  is  one  thousandth  of  the  square  root  of 
1000000  X -003249,  that  is,  of  3249. 

If  the  number  have  decimal  places,  how  do  we  proceed? 


120  ELEMENTAEY    ALGEBPtA. 

Tlnis  10.201264  would  be  dotted  10.291264  the  dot  being 
first  placed  on  the  units-figure  0;  and  the  root  will  have  one 
integral  and  three  decimal  places,  that  is,  would  be  (Ex.  3 
above)  3.208. 

If,  however,  the  given  number  be  a  decimal  of  an  odd 
number  of  places,  or  if  in  any  case  of  finding  the  square  root 
there  be  a  remainder,  then  there  is  no  exact  square  root ;  but 
we  may  approximate  to  it  as  far  as  we  please,  by  dotting,  as 
before  (rememhering  ahvays  to  set  the  dot  first  over  the  U7iits 
figure),  and  then  annexing  ciphers  (which  by  the  nature  of 
decimals  will  not  alter  the  value  of  the  number  itself),  and 
taking  them  down  as  they  are  wanted  until  we  have  got  as 
many  decimal  places  in  the  root  as  we  desire. 

Ex.  Eind  the  square  root  of  2  and  of  259.351,  to  three 
decimal  places. 


2  (1.414  &c. 
1 

24)100 
96 

259.3510  (16.104  &c, 
1 

26)159 
156 

281)400 

281 

321)335 
321 

2824)11900 
11296 

32204)141000 
128816 

Examples — 30. 
Eind  the  square  roots 

1.  Of  177241,  120409,  4816.36,  543169,  1094116,  18671041. 

2.  Of  4334724,  437.6464,  1022121,  408.8484,  16803.9369. 

3.  Extract  to  five  figures  the  square  roots  of  2.5,  2000,  .3, 

.03,  111,  .00111,  .004,  .005. 


evolutions".  121 

158.  To  find  the  cithe  root  of  a  polynomial  expression: 
AYe  know  that  the  cube  root  of  a^ -\-da'h-\-'dah'' -\-lf,  is  a  +  ^; 
and  we  shall  be  led  to  a  general  rule  for  the  extraction  of  the 
cube  root  of  any  polynomial  by  observing  the  manner  in 
which  a^-h  may  be  derived  from  a^  +  ^a^h  +  dalf  +  ^^ 

Arrange  the  terms  according  to  the  a^  +  Za^h-\-Zab'^  +  1}^ {a-\-b 

dimensions  of  one  letter,  <?-;  then  tlie  first  a^ 

term  is  a^,  and  its  cube  root  is  a,  which     3^2  \        3^2^  ^  g^^a  ^  ^s 
is  tlie  first  term  of  the  required  root,  ^d^b-\-^aJ?  -^-h^ 

Subtract  its  cube,  that  is,  a\  from  tlie  ^ 

whole  expression,  and  bring  down  the  remainder,  3a^&  +  3a5^  +  6". 
Divide  ^a^b  by  3a^,  and  the  quotient  is  Z>,  which  is  the  other  term  of  the 
required  root ;  then  form  ( 3<x'"*  +  ^ab  +  b"^)  6,  (i.  e.)  '6a^b  +  ^ab"^  +  ^^,  and  sub- 
tract it  from  the  remainder,  and  the  whole  cube  of  a  +  6  has  been  sub- 
tracted.   This  finishes  the  operation  in  the  present  case. 

If  any  quantity  be  left,  proceed  with  a  +  b  a^  a  new  a-^  its  cube,  that 
is,  c^'  +  3a'^5  +  3<3!&"-^  +  6^,  has  already  been  subtracted  from  the  proposed 
expression,  so  we  should  divide  the  remainder  hj  Z{a  +  bY  for  a  new 
term  in  the  root;  and  so  on. 

That  the  rule  may  be  thus  extended  will  be  obvious  from 
comparing  the  form  of  the  cubes  0^  a-\-h-\-c,  a  +  5-f  c  +  ^,  &c., 
with  that  oi  a  +  b,  from  which  the  rule  was  deduced. 

For, 

{a  +  b  +  cY={a  +  bf  +  ^{a  +  bYc  +  ^(a^b)c''  +  c\ 

==d'  +  {W  +  ^ah  +  b'')b+\^{a  +  bf  +  ^{a^-b)c-^c']c, 
Similarly, 
{a-^b  +  c-vdy=d'-v{Za''-^%ab  +  b'')b+[^{a  +  bY  +  ^{a  +  b)c-^c''\c 

■¥\^{a  +  b  +  cf-\-Z{a  +  b  +  c)d+d'^d; 
and  so  on. 

Pursuing  the  same  course  as  above  in  any  other  case,  if 
there  be  no  remainder,  we  conclude  that  we  have  obtained 
the  exact  cube  root. 

^x^  +  12x''y  +  Q,xy'^+y^{2x  +  y        Here  the  quantity  corresponding  to 

^x^  the  trial-divisor  ^a?  is  3  (  2x  f—  12a;\  that 

X^x^)  12x'^y  +  6xy^+y'^  to  Sd'b  is  12^'^y,  that  to  Sab""  is  6xy'',  and 

12x^y  +  6xy'^-]-y^  that  to  b^  is  y^ ;  so  that  the  w^hole  sub- 

■  trahend  is  12x^y  +  Qxy^  +  y^. 

To  find  the  cube  root  of  a  polynomial   expression. 

6 


ELEMENTARY    ALGEBRA. 


By  attending,  however,  to  the  following  hint,  the  subtrahend  mav  be 
more  easily  constructed. 


da  -f-  b    8a' 


{Sa'}-b)b 


dai  +  dab  +  W 


Sa'b  +  Sab''  +  b^ 


Wb  +  Sab''  +  W 


Set  down  first  8a,  some  little  way  to  the  left  of  the  first  remainder, 
and  then,  multiplying  this  by  a,  obtain  Sa'^  as  before ;  by  means  of  this 
trial-divisor  find  Z>,  and  annex  it  to  the  8a,  so  making  8a  +  i ;  multiply 
this  by  Z),  and  set  the  product  {Sa-\-b)b  or  Sab-\-P  under  the  3a'^,  and 
add  them  up,  making  Sa' +  'Sab  +  b'' ;  then,  multiplying  this  by  5,  we 
have  da''b-{-Sab''-{-b'\  the  quantity  required. 

The  value  of  the  above  method,  in  saving  labor,  will  be  more  fully 
seen  when  the  root  has  more  than  two  terms,  or,  if  numerical,  more 
than  two  figures.. 

Ex.  8^H  '^2x'y  +  6xy'  -hy^2x  +  y 


6x  +  y    12^;' 


-\-fjxy  +  y'' 


12X'''}' Qxy^y'' 


nx'y  +  Qxy''  +  y-' 


12x''y  +  Gxy'  +  y^ 


Examples — 31. 

Find  the  cube  roots 
1.  Ofx'  +  Cx'y  +  12xtf  +  Stj\        2.0ia'-9a'+27a-27. 
3.  Of  a;^  +  lXV+48a3+64.         4.  Of  Sa'-36a'b  +  64.ab'-27b\ 

5.  Ofa'+24:a'b  +  ld2aI)'  +  612h\ 

6.  Of  Sx'  -  Ux\j + 2^4.xtf  -  343^^ 

7.  Of  771'  -12m'nx + 4.SmnV  -  64:7i'x\ 

8.  Of  aV-loa'bx'  +  75abV -126Fx\ 

9.  Oi  a'  +  (^a'-\-lDa'-{-20a'  +  15a'  +  6a  +  l. 

10.  Ofa;"-12:?;^  +  54:r*-112a;^  +  108.'r''-48:r  +  8 

11.  Of  a'-  da'b  +  6a'b'  -  7a'b'  +  ^a'b'  -  Sab'  +  b\ 

12.  Ofa'-b'  +  c'-3{a'b-a'c-ab''-ac'-b'c  +  bc')-(jabc. 


EYOLUTIOJS^  123 

159.  The  method  of  finding  the  cube  root  of  an  algebraic 
expression  suggests  a  method  for  the  extraction  of  the  cub^ 
root  of  any  number. 

The  cube  root  of  1000  is  10 ;  the  'cube  root  of  1000000  is 
100,  and  so  on ;  -hence,  it  follows  that  the  cube  root  of  a 
number  less  than  1000  must  consist  of  only  one  figure;  the 
cube  root  of  a  number  between  1000  and  1000000,  of  two 
places  of  figures,  and  so  on. 

If,  then,  a  point  be  placed  over  every  third  figure  in  any 
number,  beginning  with  the  figure  in  the  units^  place,  the 
number  of  points  will  show  the  number  of  figures  in  the 
cube  root.  Thus,  for  example,  the  cube  root  of  405224 
consists  of  two  figures,  and  the  cube  root  of  12812904 
consists  of  three  figures. 

Suppose  the  cube  root  of  274625  required. 

180  +  5  10800  274625(60+5 

925  216000 

11725  58625 

58625 

Point  the  number  according  to  the  rule ;  thus  it  appears 
that  the  root  must  consist  of  two  places  of  figures.  Let 
a-^-h  denote  the  root,  where  a  is  the  value  of  the  figure  in 
the  tens'  place,  and  h  of  that  in  the  units'  place.  Then  a 
must  be  the  greatest  multiple  of  ten  which  has  its  cube 
less  than  274000 ;  this  is  found  to  be  60.  Place  the  cube 
of  60,  that  is  216000,  in  the  third  column  under  the  given 
number  and  subtract.  Place  three  times  60,  that  is  180, 
in  the  first  column,  and  three  times  the  square  of  60,  that 
is  10800,  in  the  second  column.  Divide  the  remainder  in 
the  third  column  by  the  number  in  the  second  column, 
that  is,  divide  58625  by  10800 ;  we  thus  obtain  5,  which 
is  the  value  of  h.     Add  5  to  the  first  column,  and  multiply 

To  find   the  cube  root  of  any  number. 


124 


ELEME^^TARY    ALGEBRA. 


the  sum  thus  formed  by  5^  that  is,  multiply  185  by  5 ;  we 
thus  obtain  925,  which  we  place  in  the  second  column  and 
add  to  the  number  already  there.  Thus  we  obtain  11725  ; 
multiply  this  by.  5,  place  the  product  in  the  third  column, 
and  subtract.  The  remainder  is  zero,  and  therefore  65  is 
the  required  cube  root. 

The  ciphers  may  be  omitted  for  brevity,  and  the  process 
will  stand  thus : 


185 


108 
925 

11725 


274625(65 
216 

58625 
58625 


It  will  be  seen  by  the  following  example,  where  the  root 
has  more  than  two  figures,  how  the  numerical  process  cor- 
responds to  the  algebraical.  The  ciphers  are  omitted,  ex- 
cept that  in  the  numbers  corresponding  to  3a^,  3a' ^  &c.,  it 
is  better  to  express  two  at  the  end:  thus 'a  is  really  4000, 
and  therefore  oa"^  is  48000000  ;  but,  as  in  the  first  remainder, 
we  only  need  the  figures  of  the  first  and  second  periods,  cor- 
responding to  43  in  the  root,  we  may  treat  the  a  as  40,  and 
thus  So"  will  be  4800,  and  3a  will  be  120,  so  that  3a +  h  will 
become  123. 


Ex. 

80677568161  (4; 

321 

64 

3^4-^  =  123  3a'^=4800 

16677 

a'  =   43 

{3a-i-b)b=   369 

a"=432 

3a'-i-3ab+F  =  ^169 

15507 

3a'  +  d=1292    3a'^  =  554700 

1170568 

{3a'-\-b)b=     2584 

da"  +  3a'b-{-I)'=657284. 

1114568 

12961     55987200 

56000161 

12961 

56000161 

56000161 

EYOLUTIOK.  .  125 

XoTE. — Our  trial-divisors  may  frequently  give  figures  too  large  for 
the  next  figure  of  the  root.  In  such  case  try  the  next  less  figure,  and 
if  necessary,  the  next  less,  until  we  get  the  right  one. 

100.  If  the  root  have  any  number  of  decimal  places,  it  is 
plain  by  the  rule  for  the  multiplication  of  decimals,  that  the 
cube  will  have  thrice  as  many ;  and  therefore  the  number  of 
decimal  places  in  every  cube  decimal  will  be  necessarily  a 
midti]jle  of  three,  and  the  number  of  decimal  places  in  the 
root  will  be  a  third  of  that  number.  Hence,  if  the  given 
cube  number  be  a  decimal,  and  consequently  have  its  num- 
ber of  decimal  places  a  multiple  of  three,  by  setting  as  be- 
fore the  dot  upon  the  units-figure,  and  then  over  every  third 
figure  on  loth  sides  of  it,  the  number  of  dots  to  the  left  will 
still  indicate  the  number  of  integral  figures  in  the  root,  and 
the  number  of  dots  to  the  right  the  number  of  decimal 
places. 

If  the  given  number  be  not  a  perfect  cube,  we  may  dot  as 
before  (always  setting  the  dot  first  upon  the  units  figure),  and 
annex  ciphers  as  in  the  case  of  the  square  root,  so  as  to  ap- 
proximate to  the  cube  root  required,  to  as  many  decimal 
places  as  we  please. 

Example.    Extract  the  cube  root  of  14102.327296. 


641 

1200 

14102.327296(24.16 

»\ 

256' 

8 

721' 

2/ 

1456 

6102 

16, 

5824 

7236 

172800 

278327 

721' 

173521 

173521 

104806296 

1 

104806296 

1742430 

0 

4341 

6 

1746 

771 

6 

126  ELEMEKTARY    ALGEBKA. 

Note. — A  careful  examination  of  the  two  columns  of  figures  on  the 
left  will  disclose  a  much  abbreviated  process  of  finding  the  divisors.  In 
the  left-hand  column,  adding  to  64,  721,  etc.,  twice  the  units-figure 
gives  the  same  result  as  multiplying  the  root  already  found  by  3.  In 
the  second  column,  adding  the  three  numbers  enclosed  by  a  brace  {i.e.^ 
the  last  true  divisor,  the  number  above  it,  and  the  square  of  the  last 
root  figure),  and  annexing  two  ciphers,  gives  the  next  trial-divisor. 

The  examples  and  explanations  above  furnish  us  the  follow- 
ing rule,  given  also  in  the  Arithmetic : 

I.  Place  a  dot  over  the  U7iits-figu're  of  the  nuwher,  and  over 
every  third  figure  to  the  left,  and  also  to  the  right  tuhen  the 
number  contains  decimals  {altvays  tahing  care  in  this  latter 
case  to  mahe  the  number  of  decimal  figures  a  multiple  of  3). 

II.  Find  the  greatest  cube  in  the  nuniber  which  forms  the 
first  period  on  the  left,  and  place  its  root  after  the  manner  of 
a  quotient  in  division.  This  root  is  the  first  figure  of  the  re- 
quired root.  Subtract  its  cube  from  the  first  period,  and  to 
the  remainder  bring  doivn  the  figures  of  the  second  period  for 
a  First  Dividend. 

III.  Multiply  the  square  of  this  first  figure  by  3,  annex 
two  ciphers,  and  find  hoio  often  this  Trial-Divisor  ^6'  con- 
tained in  the  first  dividend ;  place  the  quotient  as  the  second 
(trial)  figure  of  the  root.  Then  to  three  tiines  the  first  figure 
of  the  root  annex  this  second  figure,  and  multiply^  the  result 
by  the  second  figure ;  add  the  product  to  the  Trial-Divisor, 
and  call  the  sum  the  First  Divisor. 

IV.  Multip)ly  the  First  Divisor  by  the  second  figure  of  the 
root ;  if  the  product  be  greater  than  the  First  Dividend,  use  a 
lower  figure  for  the  second  figure  of  the  root,  and  thus  repeat 
the  process  III.  until  the  product  be  less  than  the  First  Divi- 
dend ;  subtract  this  product  from  this  dividend,  arid  to  the 
remainder  bring  doiun  the  figures  of  the  third  period  for  a 
Second  Dividend. 

Rule  for  extracting  the  cube  root  of  any  number. 


EYOLUTIOISr.  l:i? 

V.  Mulfiphf  the  square  of  the  tioo  figures  of  the  root  hy  3; 
annex  tc^o  cipher ^^,  and  proceed  as  in  III.  and  lY.  Proceed  in 
this  manner  until  all  the  'periods  have  heen  drought  dotv7i. 

iNOTii. — In  extracting  either  the  square  or  cube  root  of  any  number, 
w2ieo  a  certain  number  of  figures  in  the  root  have  been  obtained  by  the 
iommoii  rcle,  tiiat  number  ma?/  be  nearly  doubled  by  dicidoii  only. 

1.  In  the  e.xtraciion  of  the  square  root,  when  n  +  1  figures  are  found 
in  the  root,  n  more  may  be  found  by  merely  dividing  the  last  remain- 
der by  the  trial-divisor.    For,  let  N  be  tlie  number  whose  square  root  * 
is  to  be  found,  conoi&ting  of  2/i  + 1  figures. 

Let  a=  the  part  ai/eady  found  (consisting  of  n  +  1  figures,  and  n  ci- 
phers after  them,  that  i3,  altogether  of  2/2.  +  !  figures). 

Let  x-=  required  rema.uh\g  part  of  tlie  root,  consisting  of  n  figures. 


So  that 


V  j.Y~—a  +  x\ 


Then  i7  ^  vj'-'  +  2ax  +  x" ; 

J^'—a"^  x^ 

?.'.i  2a 

Now  J^^—a^  is  the  remainder,  after  n  +  1  figures  of  the  root  are  found, 

x^ 
and  2a  the  trial-divisor;  if,  then,  we  can  show  that  —  is  ^proper  frac- 

fja 

tion^  it  will  follow  that  the  integer  obtained  by  dividing  N—a^  by  2a 
will  be  X.,  the  remaining  part  of  the  root. 

But  as  X  contains  n  figures,  it  must  be  <10'^,   which  has  n  +  1  figures, 
and  x^  <W-'^ '^  and  since  a  contains  2?i  + 1  figures,  it  cannot  be  <W'^ 
(which  is  the  smallest  number  of  2n  +  l  figures). 
rj^      10-"      1 
Hence,    ,r-<^  ^^,  <^,  and  is  therefore  a  proper  fraction.     That  is,  if 
2a    2.10-'^    2  ' 

N—d^ 
the  quotient  of  —z be  taken  for  the  n  remaining  figures  of  the  root, 

the  siim  is  less  than  1. 

2.  In  the  extraction  of  the  cube  root,  when  n  +  2  figures  are  found 
in  tlie  root,  n  more  may  be  found  by  dividing  the  last  remainder  by  the 
trial-divisor. 

For  let  N=^  the  number ; 

a=  the  part  of  root  found  (consisting  of  n  +  2  figures  followed  by 
n  ciphers,  that  is,  of  2n  +  2  figures  altogether) ; 

a'=  the  required  part  of  root  (consisting  of  n  figures). 

.  Then  N=a^  +  Za^x  +  ^ax^  +  x\  and  -~--^—=.v,  +  —  +  -^  ;      and     here 

da^  a      ?ia^ 


128  ELE3IE:N^TAIiy     ALGEBEA. 

2;<10«;  and  a^  since  it  contains  2n  +  2  figures,  cannot  be   <10-"+^-, 

—  +  ^r-T,  <1.  That  is,  if  tlie  quotient  of  -tt-^t-  t)e  taken  for  the  n  re« 
a     Sa^  Sa^ 

maining  figures  of  the  root,  the  error  is  less  than  1.  Now  iV^—f^^-*— re- 
mainder after  n-\-2  figures  of  root  are  found,  and  da^  is  the  Trial- Di- 
visor for  the  next  figure.    Hence  the  rule  as  above. 


*  Examples — 32. 

Find  the  cube  roots  of 

1.  9261,  12167,   15625,   32768,   103.823,   110592,   262144, 

884.736. 

2.  1481544,  1601.613,  1953125,  1259712,  2.803221,  7077888. 

3.  Extract  to  4  figures  the  cube  roots  of  2.5,  .2,  .01,  4. 

XXII.    Simple  Equations. 

161.  The  statement  of  the  equality  of  two  algebraical 
quantities  which  differ  only  in  form,  is  called  an  "  Identity, ^^ 
An  Identity  is  true  for  any  value  whatever  of  the  letters 
which  enter  it. 

Thus,  2.'c  +  5a;=7a;;  2{a-{-x)='Za  +  2x', 

{x-\-ay=x^-\-2ax  +  a'^]  {x  +  a)  (a;— a)  =  0;^  — a',  are  Identities. 
Up  to  this  point  we  have  been  using  Identities — especially 
to  express  general  facts — by  means  of  letters.  Our  formulas 
heretofore  given  are  Identities. 

16.2.  An  equation,  however,  is  the  statement  of  the  equality 
of  two  cliff  event  algebraical  expressions;  in  which  case  the 
equality  does  not  exist  for  all  values,  but  only  for  some  par- 
ticular values,  of  one  or  more  of  the  letters  contained  in  it. 

Thus  the  equation  x—b=i,  will  be  found  true  only  when 


An  Identity.     An  Equation. 


SIMPLE   EQUATIONS.  l^/Q 

we  give  x  the  value  9;  and  x'^=z3x—2,  true  only  when  we 
give  X  the  value  1  or  2. 

In  equations,  the  question  always  is,  what  value  of  the 
letter  or  letters  not  already  known  will  verify  or  satisfy,  (i.  e.) 
make  true,  the  expressed  equality.  The  finding  of  such  value 
or  values  is  called  solving. the  equatio7i. 

163.  The  Iwo  expressions  connected  by  the  symbol  =  are 
called  sides  of  the  equation,  or  memlers  of  the  equation.  •  The 
expression  to  the  left  of  the  sign  of  equality,  is  called  the 
first  side;  and  the  expression  to  the  right  is  called  the 
second  side. 

164.  Those  quantities  to  which  particular  values  are  to  be 
given  in  order  to  satisfy  the  equation,  are  called  the  unhnoiun 
quantities.  The  last  letters  of  the  alphabet,  x,  y,  z,  &c.,  are 
usually  employed  to  denote  these  quantities. 

165.  An  equation  is  said  to  be  satisfied  by  any  value  of  the 
unknown  quantity  which  makes  the  values  of  the  two  sides 
of  the  equation  the  same,  {i.  e.)  which  makes  the  Equation  an 
Identity. 

This  includes  the  case  where  all  the  terms  of  an  equation 
lie  on  one  side  and  0  on  the  other,  as  in  x^—^x  +  '^  —  Q,  which 
is  satisfied  by  1  or  2,  either  of  which  being  put  for  x  makes 
the  first  side  also  0. 

Those  values  of  the  unknown  quantities  by  which  the 
equation  is  satisfied,  are  called  the  roots  of  the  equation. 

Thus, '7  is  the  root  of  x—d=z4:;  1  and  2  are  the  roots 
ofx'-dx  +  'Z^O. 

168.  An  equation  of  one  unknown  quantity,  when  cleared 
of  surds  and  fractions,  is  said  to  be  of  as  many  dimensions  as 
there  are  units  in  the  index  of  the  highest  power  of  the  un- 
known quantity.  1^0 

Thus,  cc— 5=4  is  an  equation  of   one  dimension,  or,  of 

Solving  the  Equation.    Sides  or  Members  of  the  Equation.     Unknown  Qua?: 
titles.    Satisfying  an  Equation. 


130  ELEMEKTARY    ALGEBRA. 

the  first  degree,  or  a  simple  equation;  x^  =  'dx—^,  is  of 
tiuo  dimensions,  or,  of  the  second  degree,  or  a  quad- 
ratic equation;  x^  —  h^=.^oi^  is  of  three  dimensions,  or  of 
the  tliird  degree,  or  a  cuhic  equation;  x^  —  ^x^=.V6,  is  of 
four  dimensions,  or  the  fourth  degree,  or  a  hiquadratic 
equation,  &c.,  &c. 

167.  In  the  present  chapter  we  shall  show  how  to  solye 
simple  equations.  We  have  first  to  indicate  some  operations 
which  we  may  perform  on  any  equation  without  destroying 
the  equality  which  it  expresses. 

168.  If  every  term  of  each  side  of  an  equation  he  multiplied 
ly  the  same  quantity,  the  ttvo  sides  ivill  still  he  equal. 

For,  if  equals  he  multiplied  by  the  same  quantity,  the 
results  are  equal. 

This  principle  is  chiefly  used  for  clearing  an  equation  of 
fractions,  if  they  stand  in  the  way  of  solving  it. 

Thus,  taking  the  equation  ^x—^  —  -^,  multiplying  every 

o 

term  by  3,  the  denominator  of  the  fractional  term,  we  have 

21:?;  — 18=3X-^,  or  21:^^—18  =  5^,  in  which  no  fraction  ap- 
o 

pears. 

An  equation  of  several  fractional  terms  may  be  cleared  of 

fractions  by  multiplying  every  term  by  any  common  multiple 

of  all  the  denominators.     If  the  L.o.M.of  the  denominators  be 

employed,  the  equation  will  be  expressed  in  its  simplest  terms. 

Take,  for  example,  —  +  —  +  —  =  9. 

o         4         D 

Multiply  every  term  by  3  X  4  X  6,  or,  72 ;  thus, 

72a;     72a;      72a;         '^^ 

that  is,  24a;  +  18a:+12a:=648,  cleared  effractions. 

RootR  of  an  Equation.    Difi'firont  kinds  of  Equations. 


SIMPLE   EQUATION'S.  131 

Instead  of  multiplying  every  term  by  72,  we  may  multiply 
eyery  term   by  12,  the  l.c.m.  of   3,   4,  and  6.      We  would 

\2x     VHx     1.2x 

thus  liaye  — — H — --  +  —-  =  108;  that  is,  4tx  +  dx+2x=108, 

O  ~c  K) 

expressed  in  simple  terms. 

-^        ^T        ,.  ,.       6x  +  4:     7x  +  5     28     x—1     „^ 

hx.     Clear  the  equation  — ttt"— ~^~~o~"^  ^^  ™^'" 

/C  10         o         /^ 

tions. 

The  L.C.M.  of  the  denominators  is  10.     Multiply  by  10. 

Thus,  6{6x  +  4:)-{7x  +  5)  =  b6-6{x-l); 

that  is,  26x-{-20^'7x-6=z66-6x  +  6. 

The  beginner  should  write  the  operations  out  in  full,  as 

above,  using  brackets,  in  order  that  he  may  attend  to  the 

.       .  .  7.T  +  5         ,       x—1 

signs  01  such  expressions  as — r— ,  and  —  . 

10  Zi 

169.  Any  term  may  he  transferred  from  one  side  of  a7i 
equation  to  the  otlter  side  ivitliout  destroying  the  equality,  j^ro- 
vided  2ve  change  the  sign  of  the  term. 

This  transference  is  called  transposing. 
Suppose,  for  example,  x—a  —  h—y. 

Add  a  to  each  side  (which  of  course  will  not  destroy  the 
equality) ; 

then,  x—a-\-a—'b—y-^a\ 

that  is,  x='b—y-\-a. 

NoAV  subtract  h  from  each  side ;  thus, 

x—'b—lj^a—y—'b\ 

that  is,  x—t^za—y. 

Here  we  see  that  —a  has  been  removed  from  one  side  of 
the  equation,  and  appears  as  +a  on  the  other  side;  and  -\-'b 
has  been  removed  from  one  side  and  appears  as  —  Z>  on  the 
other  side. 

Transposition  of  Terms. 


132  eleme:n^tary  algebra. 

170.  If  the  sign  of  every  term  of  an  equation  he  changed  the 
equality  still  holds. 

This  follows  from  Art.  169,  by  transposing  every  term. 
Thus,  suppose,  for  example,  that  x—a=t)—y. 

By  transposition,        y—h—a—x'^ 

that  is,  a—x=:y—ib. 

And  this  result  is  what  we  shall  obtain  if  we  change  the  sign 
of  every  term  in  the  original  equation. 

It  is  also  clear  that  if  the  same  quantity  occur  with  the 
same  sign  on  both  sides  of  the  equation,  it  may  he  erased 
from  hoth  sides.  For,  by  the  erasure  w^e  either  subtract 
equals  from  equals  or  add  equals  to  equals,  according  as  the 
sign  of  the  term  erased  is  +  or  —. 

171.  Every  term  of  each  side  of  an  equation  may  he  divided 
hy  the  same  quantity  luithout  destroying  the  equality  expressed 
hy  it. 

For,  if  equals  be  divided  by  the  same  qua;ntity,  the  quotients 
are  equal. 

Thus,  the  equation  12a; +  5.^=1 3 6,  or  17a?  =  13  6,  gives 

17^_136  _1^_Q 

17  —  1 7  '  ^^  '^ —  17  — 

..       .n  -,  ct^      ^  'b 

Ai^o,\iax=o.        —  =  — ,  or  x=:—, 
•  a      a  a 

Again:   if  24:?;+18:?;  +  12:^:  =  648  be  divided  by  6,  we  get 

4a;+32:  +  2a-108,  or  92;=108. 

9ix     108  108     ,^ 

Hence,  ~a—~K~^  ^"^  x——=V^,     ^ 


then 


Again:  M ax-\-hx—cx^^d^(dY  (a-\-h--c)x—d*y 
{a-\'h—c)x  d  _       ^ 


a-\-h—c       a  +  h  —  c^  a-\-h- 


Chaflge  of  the  signs  of  an  equation.    Division  of  the  terms  of  an  equation. 


SIMPLE    EQUATIOXS.  133 

172.  The  operations  indicated  in  the  preceding  articles 
may  be  performed  upon  equations  of  any  degree,  and  con- 
taining any  number  of  unknowns ;  for  they  depend  on 
principles  true  of  every  equation.  Of  course  all  these  opera- 
tions can  be  performed  on  Identities,  or  identical  equations, 
as  they  are  sometimes  called. 

173.  To  solve  a  simple  equation  of  one  unknown  quantity : 

Eule.  Clear  the  equation  of  fractions,  if  necessary.  Collect 
all  the  terms  involving  the  unknown  quantity  on  one  side  of 
the  equation  and  the  known  quantities  on  the  other  side,  tra^is- 
posing  them,  when  necessary,  with  change  of  sign.  Add  to- 
gether the  terms  of  each  side,  and  divide  both  sides  dy  the  coef- 
ficient or  sum  of  the  coefficients  of  the  unknoiun  quantity  ;  and 
thus  the  root  required  will  he  found. 

Note  I. — Erase  terms  by  Art.  170,  or  simpUfy  the  equation  by  di- 
vision, Art.  171,  at  any  stage  of  the  process. 

Note  II. — It  is  usual  to  collect  all  the  unknown  quantities  on  the 
first  side,  and  the  known  on  the  second  side  of  the  equation. 

174.  We  shall  now  give  some  examples. 
Ex.  1.     Solve  7:?;  +  25  =  35  +  5.T. 

Here  there  are  no  fractions;  by  transposing  we  have 

7:2;-5:z:=35-25; 
that  is,  2<c=10; 

dividing  by  2,  x=—-  =  o. 

We  may  verify  this  result  by  putting  5  for  x  in  the'  original 
equation ;  then  each  side  is  equal  to  60. 

Ex.  2.     4.x  +  b  =  10x-U. 

Here       Wx-4.x=b-\-l(j;  ,\  Qx=2l,  ^ndi  x^zz^ z=z^=^, 

Ex.  3.     b{x-\-l)-2  =  ^{x-b). 

Tlule  for  solving  a  Simple  Equation.    Note  I.    Note  II. 


134  ELEMENTARY     ALGEBRA. 

Here^  remoying  the  brackets,  6x+6—2  =  dx—16; 

.-.  5x—3x=:—16—6-i-2,  or  2^=— 18,  and  /.  ^=— 9. 

Ex.  4.     Solye4:{3x--2)-2{4.x-3)-3{4:-x)=0. 

Performing  the  multiplications  indicated, 

12x-8-{Sx-6)-{12-3x)=0. 

Eemoving  the  brackets, 

12a;-,8— 82:  + 6— 12  +  3^=0; 

collecting  the  terms,  7a;— 14=0; 

transposing,  7^==  14 ; 

14 
dividing  by  7,  x=—=2. 

The  student  will  find  it  a  useful  exercise  to  yerify  the  cor- 
rectness of  his  solutions.  Thus,  in  the  aboye  example,  if  we 
put  2  for  X  in  the  original  equation,  we  shall  obtain  16  —  10—6, 
that  is  0,  as  it  should  be. 

Ex.  5.  dx-^2x—a=3x-{-2c. 

Transposing,  I)X  +  2x—dx=a  +  2c; 

or,  dx—x=a  +  2c; 

collecting  the  coefficients,    (^—1)  x=a  +  2c; 

a -{-2c 


dividing  by  Z>  — 1, 


b-1 


Examples — 33. 
1.  4.x-2  =  3x-{-3.     2.  3^  +  7=:9a;-5.      3.  4.x-\-9=zSx-3, 
4.  3-{-2x=7—6x.     5.  x—7  +  16x.  6.  7nx-{-a  =  nx-{-d. 

7.  3{:?:-2)+4=4(3-.t).      8.  5-3  (4-^-) +4  (3-22:)r=0.  -J 
9.  13:i;-21  (a;-3)  =  10-21  (3-a;). 
10.  6{a  +  x)-2x  =  3{a-5x), 


SIMPLE    EQUATIOXS.  '  135 

11.  3{x-3)-2{x-2)+x-l:=x-{-3  +  2{x  +  2)  +  3{x+l). 

12.  2.i;-l-2  (3:?;"- 2)  +3  (4a;-3)-4  (5:?;-4):=0. 

13.  {2-{-x){a-3)  =  -4.-2ax. 

14.  {7)1 +n)  {m—x)=7n  {?i—x). 

15.  5.'?;-[8a:-3{16-6a;-(4-5^)}]=:6. 

175.  The  following  examples  will  illustrate  the  solution  of 
equations  containing  fractions. 

Ex.1.     If  |-|:- 1=^-3,  find:.. 

Multiplying  by  2  X  3,  or  6, 

3:?;-10a;-8  =8x-lS; 

transposing,  3.^—10:^—82:=:  8  —18; 

combining,  — 15a;=— 10; 

dividing  by —15,  x= — — =— . 

—  J  0       o 

Ex.  2.     ix-^x+lx=ll+^x. 

Here  we  first  clear  the  equation  of  fractions,  by  multiply 
ing  every  term  by  24,  the  l.c.m.  of  the  denominators,  and 
(observing  that  in  the  first  fraction  ^^  =  12,  in  the  second  ^^ 
=8,  and  so  in  the  others)  thus  we  get  12:?;— 8x2:^:+6x3a; 
=  264  + 3a:,  or 

12x-iex-\-lSx=264:-{-dx; 
.-.  12.i;-16r^  +  18.T-3.T=264;  .-.  lla;=:264,  and  ^^.-^^V^^^- 
Ex.  3.     If  x  +  —^:=:12-—,:p^,  find  X. 
To  clear  of  fractions,  multiply  by  2x3,  or  6,  and  we  have 

62^  +  3  (32:-5)=.72-2  (2x-4) ; 
or  62;+(9a;-15)  =  72-(42:-8); 
.-.  C):j^  +  9.7:-15  =  72-4.t4-8. 


136  ELEMEKTARY     ALGEBRA. 

Transposing,       6x-{-9x  +  4:X=l!2-{-8-{-16; 
combining,  192;=  95; 

95 
dividing,  0:=— =5. 

Ex.  4.     Solve  h6x+3)-^{16-5x)=31!-4.x. 

This  is  the  same  as 

6x^-3     16-5x    ^^     , 
—- ^ — =37-4:X. 

Multiplying  by  21,  7  {6x  +  3)-'d  (16-52;)  =  21  (37-42;) 
that  is,  352;  +  21-48  +  152;=r777-842;; 

transposing,  352^  +  152; +  842;=  777 -21 +  48; 

that  is,  1342;  =  804; 

therefore,  ^= ttt:  =  6. 

134 

Ex.5.     i{x-\-l)+i{x-^2)=16-i{x  +  3). 

Multiplying  by  12,  we  have 

6(2;  +  l)+4(2;  +  2)  =  192-3(2;+3), 

or    62;+ 6   +  42;  + 8  =192  — 32;— 9; 

.•.62;+42;  +  32;=192-9-6-8;   .•.132;=169,  and  x=\%^z=:13. 

Examples— 34. 
1.  ^2;+-j2;=2;— 7.  2.  ix—^x=lx—l. 

0  o 

2r     (T 2 

5.  y+-^=a;-4  6.  i(9-3a;)=|-TV(7a.-  18). 

7.  x+\{U-x)=^{%l-x).  8;  2a;-^=|(3-2a;)+ia;. 

9.  ^(2a:+7)-yT(9.'P-8)=i(a;-ll). 


SIMPLE  equatio:n"S.  137 

,^   x-a     2x—3b     a—x    ^     ^^    6x—7     2x+'7     .. 

-,      a;-2    «— 3     „  ,„    x+3     x+4:    x+5     ,„ 

13.  ^_i_  -^+^=0.         13.  _  +  _+_=:16. 

14.  !^=7  +  .-^-i^i.  15.  ^i-^i.£^. 

3a;-5     5a;-3     „„     „     ,„    7:r-4    „„     4-7a;  7 

12' 


IC.  -3 ^+2|=0.    17.  ^^-  +  3|  +  -^=a;-^. 


,„    3— a;    3— a;    4—2;    5— a;     3 
18.— +  —  +  —+— +  ^=0. 

,„   5a;-3     9-a;    5x    19, 


XXIII.  -  Simple  Equations  CoNTrNUED. 

176.  We  shall  now  give  some  examples  which  are  a  little 
more  difficult  than  those  in  the  preceding  chapter.  In  many 
of  these  examples  the  common  multiple  of  all  the  denomina- 
tors is  too  large  to  be  conveniently  employed.  In  such  a  case 
we  may  see  whether  two  or  three  of  the  denominators  have  a 
simple  common  multiple,  and  get  rid  of  these  fractions  first, 
observing  to  collect  the  terms  and  simplify  as  much  as  pos- 
sible after  each  step. 

^      ^     2.^  +  3     :i;— 12     32:4-1     ^,  .  4:X  +  3 

^-^-  ^-   ^T 3-+-^=^*+-]^- 

Here  the  l.c.m.  of  all  the  denominators  would  be  132 ;  but 
as  12  will  include  three  of  them,  multiplying  by  it  (having 
iirst  changed  5^  to  ^3^),  we  get 

^^?-^tl)-4(a;-12)  +  3(3a;  +  l)  =  64  +  4a;+3; 
.\\^{2x-^d)-4cX+4:8  +  9x  +  3  =  64:+4.X'\-3; 

When  the  Common   Multiple    of  all    the    denominators    is    too  large  for  coa 
venience,  what  course  may  you  pursue? 


138  ELEMENTARY    ALGEBRA. 

hence,  collecting  terms  and  simplifying,  we  liaye 
m2x+d)-4.x-{-9x-4:X=64.-\-3-4:8-3, 
ovii{2x  +  3)+x=:16; 
.-.  12  (2:2;+3) +11^=176,  or  24^;  + 112;= 176 -36; 
/.  o6x=:^14:0,  and  2;=V¥=^- 

177.  It  will  often  happen  that  the  icnlcnoion  quantity  is 
found  in  the  denominator  of  one  or  more  of  the  fractions. 

x^ooi  24352 

Ex.2.    Solye       —+—=_-] _. 

X       X       x       X       11 

Since  the  first  four  fractions  haye  a  common  denominator, 

by  addition,  _=,___; 

8      6      2 
transposmg,  ^~"^""l7' 

2      2 
combining,  '^=17' 

.-.  a;  =17. 

Multiplying  by  3^;,      9  —  2  =  5  + :?; ;       :.  x=2. 

If  any  of  the  denominators  which  contain  the  unknown 
quantity  consists  of  two  or  more  terms,  it  will  generally  be 
adyisable  to  follow  the  method  of  Art.  176,  and  clear  the 
equation  of  the  simplest  denominators  first,  leaying  tho 
others  to  be  dealt  with  afterward,  when,  by  transposing, 
collecting  the  terms,  &c.,  the  equation  has  been  reduced  to 
feiver  terms.  Or,  if  all  the  denominators  consist  of  two  or 
more  terms,  then  they  may  be  cleared  off  singly,  one  hy  cne, 
till  all  haye  disappeared. 

How  may  you  proceed  when  the  unknown   quantity  is  found  in   the  denora 
Inator?    When  any  of  the  denominators  coneists  of  more  than  two  terms? 


Ex.  4.    Solve 


SIMPLE   EQUATIONS.  139 

6x-\-13      3.T+5      2x 


15         5:^;— 25      5 
Multiply  by  15  to  clear  away  the  simjyie  denominators  first, 
and  we  have 


.,  .        ^^     15(3:^4-5) 

y,  and  transposing,  13=      \  ^^    ? 


erasing,  ana  transposing,  16=    k   _<^- 

or,  dividing  numerator  and  denominator  of  the  fraction  by  5, 

^^^3(3a;  +  5)^ 
x—b 

Multiplying  by  a;— 5,  13:?;— 65  =  9^;  + 15  ; 

transposing,  13^;— 9^=65  +  15 ; 

combining,  4^ = 80 ; 

80 
dividing,  x=——20, 

Ex.5.     Solve— g ___^_-_. 

To  remove  first  the  denominators  18  and  9,  multiply  the 
whole  by  18,  and  we  have 

10^_,17_-j^_^.10:.-8; 

erasing,  and  transposing, 

216r.+36. 

,  .    .  ^^     2162:  +  36 

combining,  2o=-— r— ; 

Wx — o 

multiplying  by  1Lt-8,  25  (ll:r— 8)=216:?:  +  36; 

or,  275:^:-200=:216a;  +  36; 

transposing,  275a;— 2l6x==200  +  36; 

combining,  5  9a;  rr:  23  6 ; 

dividing,  0;:=— ^=4. 


140  ELEMENTARY    ALGEBRA. 

-^      ^      ^  ,       2x-hS     ^x  +  6     3x  +  d 
JbiX.  6.    'bolve  — r^^z 7  +  ?; T. 

x+1     4.X  +  4:    Sx  +  l 

Here  it  is  convenient  to  multiply  by  ^x-\-4:,  that  is  bv 
4(0^+1); 

.  /^       ox      .       .  .  4  (i?; 4- 1)3(^+1) 
thus  4(2ic  +  3)  =  4a;+5+-^-— -^^--^--^; 

12  (a^  +  l)'  ^ 


therefore,  8a; +  12— 4^— 5  = 

that  is^  42^+7= 


12  {x-\-iy 


dx-\-l 

Multiplying  by  3a;  +  l,  (3:2:  +  l)  (42^+7)  =  12  (a:+l)'; 

that  is,  12:c'+25^  +  7=12a;'+24x+12. 

Here  l^x^  is  found  on  both  sides  of  the  equation.  Ee- 
move  it  by  subtraction,  and  the  equation  becomes  a  sim2)le 
equation;  that  is, 

25^  +  7=24:^;-fl2; 

or,  252J— 24^=12-7; 


Ex.  7.     Solve 


x—1     x—2     x—4:     x—5 


x—2     x—3     x—5     x—6' 

Eeducing  the  terms  on  the  first  side  to  a  common  denom- 

.     ^  ^  x^~Ax  +  3-(x'-4:X  +  4.)  1 

mator,  we  get ^, — ,  or  —  -, -^, — -^, 

"  {x—2)  [x—d)  {x^2)  (x—d) 

Eeducing  the  terms  of  the   second  side  to  a  common  de- 
nominator, we  have  for  this  side 

.^'-rl0:c  +  24-(.'r'-10:r+25)  1 


(x-b)(x-Q>)  '  (x-b){x-Qy 

Thus  the  proposed  equation  becomes 

1 1^ ^ 

""  {x-2)  {x-^)"      (x-5)  (.T-6)  ' 


SIMPLE   EQUATIONS.  141 

1  1 


Changing  the  Signs,  l^_^)j^_^^-^',j(^^Ze)^ 
clearing  of  fractions,  (x— 5)  (x—Q)  =  {x—2)  0^—3) ; 
that  is,  x^--llx  +  30=x''—6x  +  6; 

whence,  —  lice  +  5a;  =6— 30; 

that  is,  —  6.T  =  —  24 ; 

therefore,  a: =4. 

A5x-,76     1.2     .dx-.6 


Ex.  8.    Solve  .5^;+- 


.6        ~  .2  .9 


To  insure  accuracy,  it  is  advisable  to  express  all  the  deci- 
mals as  common  fractions ;  thus 

6x    10/45^  __75^\_10     l^_10/3^_j6\ 

Simplifying,  |  +|(|-|)  -=6-  (-|-|-)  ; 

,,    ,  .  a;     3cc     5      ^     a;      2 

that  IS,  _  +  _^_:.6--+-. 

Multiplying  by  12,     6x+9x-16=:'72-4:X  +  S  ; 
transposing,  19a;=72+8  +  15  =  95; 

therefore,  x=—=5. 

178.  Complex  fractions  in  an  equation  should  first  be  re- 
duced to  simple  ones,  by  the  known  rules. 

T,      ^       25-4:?;     16.^  +  44     .       23 

^"^•'-    -^+-3-^+?=^+^T 

Hei  e,  first  simplifying  the  complex  fractions,  we  get 

'75-x        80:r  +  21  _       _23_^ 
3(a;+l)"^5(32:  +  2)~    ^^+1' 

Complex  fractions  in  an  equation. 


142  ELEMEKTARY    ALGEBRxV. 

,^.  ,  .       ^    ,.    375-5.-?;     240:^+63     ^.  ,   345 
then,  multiplying  by  lo, ——  +  -———— =  7o  + 


x  +  1    ^    32^+2      '         x  +  1' 
whence,  multiplying  hj  x+1, 

o^.     r.    .  2402;'+303.T+63     ,,^       ^.  .  o... 
875— 5:^;^ ^-— -^ =75iz;  + 75+345; 

OX  -J-  /i 

or,  simplifying, 

J— — r =  752;+5aj  +  75+345— 375=80:^+45: 

3a;+2 

/.  240^;'  + 303:^ +  63  =  240;^'+ 295a; +  90,  and  8a;:=27,  or  x=3%. 

179.  We  will  now  solve  three  more  equations,  in  whi^'^ 
letters  are  used  to  represent  known  quantities. 

Ex.  10.    Solye  -  +  ^=^. 
a      0 

Multiplying  by  ^Z>,         Ix-^-ax—dbc^ 

that  is,  {a-^'b)x=dbc\ 

dbc 

,\  x=-—-. 
a-\-o 

Ex.  11.    Solve  a— ^ — yd —X, 

0  a 

Multiplying  by  ab,    a^ {x—d)-\- If  (x—l)  =  abx ; 

that  is,  a'^x—a^-\-l'^x—lf=:abx] 

transposing  and  collecting,  a^x-\-lfx—abx—a^  +  V ; 

that  is,  {d^—ah-\-¥^  x^cc" -\-W  \ 

dividing  by  c^-\-db-\-  Z>^  x  =  ,_  ^ ; 

.•.  x=a+b, 

-n      -.n      CI  1       ^— ^     (2x~ay 

Ex.  12.     Solve  7=77^ r(?. 

x—o     {2x—o) 

Clearing  of  fractions, 

(x-a)  {2x-hy  =  {x-b)  {2x-ay; 
that  is,       (x-a)  {^x''-ihx-^b')=z{x-b)  (4a;'-4«a;  +  a'), 


SIMPLE   EQUATIONS.  J  43 


multiplying,  we  obtain 

4.x'-4.x\a^h)^x{4.a'b-\-l)'')-ah'' 
=  4a;' - 4a;' {a -\-i)-\-x  {4.ah + a')  - a'Z> ; 
whence,  lfx—ah'^=a^x^a^h\ 

/.  {a'—V^)  x—a^l—aV—ab  (a—V) ; 
__ah{a—l)_  ah 


Examples — 35. 

12     _1__29  _iL__££_  3      l^B   _   216 

'^"^12a;~24*  '  a;-2~a;-3*  '  3a;-4~5a;-6' 

2^T3~4^^'       ^'       2     ~     3     "^    4        6~^'^  ' 

1  3 

--a;— 3     -a;— 10     .  ^_ 

2  ,4  ,  4:—x    10— a; 


5  2  4  6 


5a;~8     ^     3a;-8 


.    a;-2  ,  1  2a;-l  ^.      ^  ^     a;^+3     ^ 

9.  _+_  =  ^____.  10.  a;+l-^^=2. 

a;-l_7a;-21  7a;-4    7a;-26 

*  a;-2~7a;-2G'  *   a;-l  ~"  x-3  ' 

X     dx     71_3a;  +  l  .  .    2a;-6_2a;— 5 

7  ~  2  "^  7  ~     2     "^  ■^^*  3a;-8~3a;-7' 

15.  a;--3~(3-a;)(a;  +  l)=:a;(a;-3)+8. 

IG.  3-a;-2(a;-l)  (a;  +  2)^(aj-3)  (5-2a;). 

17.  I+^_i+^=,7^^.  18.  (a;  +  7)(a:+l)  =  (a-4-3r. 


144  ELEMEKTARY    ALGEBRA. 

19.  ^{2x-10)-^{3x-4.0)  =  1d-^{d^-9:). 

2x+l       ^+12  ~ 

21.  ^i^{2x-3)-i{dx-2)=i{^x-3)-d^^^. 

22.  5(^__9)+_T(^_5)^9(^_7)  +  l|. 

23.  ,-V(2^-l)-3J^(3^-2)=^V(^-12)-^VG^+12). 

24.  ^(^x+20)-^\{3x  +  4.)=^^-^{3x  +  l)-M^9-Sx). 
^^    dx-1      4x-2     1  ,,^        2  1  6 

20.  rr T  —  n 7^  =  7r-  ^6. 5  + 


27. 


2:^-1      3:z;-2     6*  '  22;-3^a;-2     3:z;  +  2' 

x—4:    x—6     x—7     x—8 


x—b     x—^     x—d>    x—9 


28.  JL,  +  ^=^  +  ^' 


x—2     x—l     x—1     x—6 

29.  ^-l±^.-  ~=:1.     30.  ,6x-2=--.26x+,2x-l. 
3 — X      2  —  x      l-\-x 

31.  .5:z;  +  .6a3-.8=:.75.'?;  +  .25. 

.135aj-.225     .36      Mx-.IS 


32.  .15a;  +  - 


.6         ~.2  .9       • 

fl^— a:     ^  J+a:  ^,    x^  —  a""      a-x    2x    a 

33.  a~^f 0 =x,  34.  —j tt—t • 

h  a  ox  0        0     X 

35.  x{x—a)-{-x{x  —  'b)=2{x—a){x—h). 

36.  ix-a){.-l)  =  {.-a-iy.     37.  ^-^j=^. 

28  -1 ^=-^^. 

*  x—a      x—h     x^  —  ab 


39. 


40. 


SIMPLE   EQUATIOJS'S.  145 

111 


x—a      x—a-\-c    x—b  —  o      x—V 
m  X  —a  —  h     mx  —a — c 


nx—c~d     nx—h—cV 
41.  (ci-l)  {x-c)-{b-c)  {x-a)-{c-a)  {x~b)=:0. 

x—a      x  +  a      2ax 


42. 


a  —  b       a  +  b    a^  —  b''' 


,_       x—a         x—a—X        x—b         x—b—1       /rj     t^     ^ 

43. r -= J — ~ r~^.     (See  Ex.  6, 

x—a—1      x—a— 2    x—b—1      x—b— 2       ^ 

Art.  177.) 

XXIV.    Peoblems  solved  by  Simple  Equations. 

180.  We  shall  now  see  tlie  practical  application  of  the  above 
methods  in  the  solution  of  many  arithmetical  problems.  In 
these  problems  certain  quantities  are  given,  and  another 
quantity,  Avhich  has  certain  given  relations  to  these,  has  to 
be  found :  the  quantity  which  has  to  be  found  is  called  the 
unhioivn  quantity.  The  relations  between  the  given  quanti- 
ties and  this  unknoiun  are  expressed  in  the  enunciation  of  the 
problem  in  ordinary  language,  -and  these  are  to  be  translated 
into  algebraical  expressions,  to  be  used  in  the  solution  of  the 
problem.  The  method  of  solving  the  problem  may  be  given 
in  general  terms,  as  folloAvs : 

Put  X  to  represent  the  unhnoion  quantity.  Set  doivn,  in 
algebraical  language,  the  statements  made  in  the  problem,  and 
the  relations  betiueen  the  unhnoiun  quantity  and  given  quanti- 
ties derived  from  these  statements,  using  x  tvhenever  the  un- 
hnoion quayitity  occurs.  We  shall  thus  arrive  at  an  equation 
from  luhich  the  value  ofx  may  be  found. 

Solution  of  problems  by  means  of  Simple  Equations.    State  in  general  terms 
the  steps   to  be  taken. 

7 


146  ele:>ii:xtaky   algekha. 

Ex.  1.  What  number  is  that,  to  which  if  8  be  added,  one- 
fourth  of  the  sum  is  equal  to  29  ? 

Let  X  represent  the  number  required. 

Adding  8  to  it,  we  have  x  +  8,  and  one-fourth  of  this  is  \  {x  4-  8) ; 

Ave  have,  therefore,  the  equation  i(a;  +  8)=29;  whence  it;  =  108. 

Ex.  2.  What  number  is  that,  the  double  of  which  exceeds 
its  half  by  6  ? 

Let  a'=  the  number ;  then  the  double  of  x  is  2.r,  and  the  half  of  x  is 
\X',  hence,  2x—\x=^%\  whence  a;=4. 

Ex.  3.  The  ages  of  3  children  together  amount  to  24 
years,  and  they  were  born  two  years  apart :  what  is  the  age 
of  each  ? 


Here  we  have 

Known  quantities. 

1.  The  sum  of  the  ages  of  all 
three,  24  years. 

2.  The  difference  between  the 
ages  of  any  two  of  them. 


Unknown  and  required. 

1.  Age  of  youngest. 

2.  Age  of  next. 

3.  Age  of  the  oldest. 


But,  in  reality,  we  have  only  one  unknown  quantity  to  find^  because, 
when  we  know  the  age  of  one  of  the  children,  the  ages  of  the  two 
others  immediately  follow.     So  that  we  say, 

let  x  be  the  age  of  the  youngest ; 

then  X  +  2= next ; 

and  «  +  4= oldest. 

Thus  far  we  have  expressed,  algebraically,  07ie  of  the  two  known 
coftditions  of  the  problem.  There  still  remains  to  notice,  that  the  sum 
of  the  ages  is  24  years.  Now  this  sum  is  ^x  +  6,  adding  together  x^  a;+2, 
and  x-\-^; 

:.  3.-??  +  6=24,  an  equation  from  which  to  find  x. 

Transposing,  3a=24— 6,  or  18; 

dividing,  ^="5" »  ^^  ^' 

o 

.'.  the  age  of  the  youngest  is  6  years, 

next     . .    8  . 

oldest   . .  10 


SIMPLE   EQUAT]0]S'S.  147 

Ex.  4.  A  cask,  wliicli  held  270  gallons,  was  filled  with  a 
mixture  of  brandy,  wine,  and  water.  There  were  30  gallons 
of  wine  in  it  more  than  of  brandy,  and  30  of  water  more 
than  there  were  of  wine  and  brandy  together.  How  manv 
were  there  of  each  ? 

Let        ^=:  number  of  gals,  of  brandy  ; 

.*.  x  +  dO= wine; 

and2aj  +  30= wine  and  brandy  together; 

...  2a,'  +  30  +  30  or  2x  +  60=gals.  of  water ; 
but  the  whole  number  of  gallons  was  270  ; 

.-.  a^  +  (aj  +  30)  +  (2,2;  +  60)=270; 
whence  x=  45,  the  number  of  gals,  of  brandy, 

25  +  30=  75, wine, 

22^  +  60=150, water. 

Ex.  5.  A  sum  of  £50  is  to  be  divided  among  A,  B,.and  C, 
so  that  A  may  have  13  guineas  more  than  B,  and  C  £5  more 
than  A  :  determine  their  shares. 

Tjet  x=B^s  share  in  sliilUngs  : 

.'.  «  +  273=^'s,  and  .(;r  + 273) +  100  or  ^  +  373=6''s  ; 

.-.,  since  £50=1000s.,  {x  +  27S)  +  x  +  {x  +  d7S)=Sx  +  64.Q=1000; 

.-.  3^r=354,  and  0^=118,  :c  +  273=391,  aj  +  373=491, 

and  the  shares  are  391s.,  118s.,  491s.,  or  £19  lis.,  £5  18s.,  £24  lis.,  re- 
spectively. 

Ex.  6.   A,  B,  C  divide  among  themselves  620  cartridges,  A 
taking  4  to  ^'s  3,  and  6  to  (7's  5 :  how  many  did  each  take  ? 
Let  .Tr=^'s  share ;  then  |a;=jB's,  ■|.c=(7's ; 

.-.  aj  +  |aj+ 5,^=620;  whence  ir=240,  |.t'=180,  |-a^=200. 

We  might  have  avoided  fractions  by  assuming  12x  for  J.'s  share,  whea 
we  should  have  had  9aj=^'s,  and  10.2^=:  C^  ; 

.'.  12a^  +  9^  +  10i=620 ;  whence  a'=20 ; 

and  the  shares  are  240,  180,  200,  as  before. 


148  ELEMEXTARY     xVLGEBRA. 

Ex.  7.  A  line  is  2  feet  4  inches  long ;  it  is  required  to  di- 
vide it  into  two  parts,  such  that  one  part  may  be  three- 
fourths  of  the  other  part. 

Let  X  denote  the  number  of  inches  in  the  larger  part ;  then  —  will 
denote  the  number  of  inches  in  the  other  part. 

The  number  of  inches  in' the  whole  line  is  28  ;  therefore 

.  +  1-28; 

Whence,  ^  +  3a?=112  ; 

that  is,  7^=112 ; 

and,  x=10. 

Thus  one  part  is  16  inches  long,  and  the  other  part  12  inches  long. 

Ex.  8.  A  grocer  has  some  tea  worth  half-a-dollar  a  lb.,  and 
some  worth  87^  cents  a  lb. ;  how  many  lbs.  must  he  take 
of  each  sort  to  produce  100  lbs.  of  a  mixture  worth  62^ 
cents  a  lb.  ? 

Let  x=  the  number  of  pounds  of  the  first  sort;  then  100— a?  will  de 
note  the  number  of  lbs.  of  the  second  sort.  The  value  of  the  x  lbs.  is 
ix  dollars,  and  the  value  of  the  (100— 3j)  lbs.  is  KlOO— .r)  dollars;  and 
the  whole  value  is  to  be  |  X 100  dollars. 

Therefore,  §  X 100= Jc  + 1(100 -x) ; 

multiplying  by  8,  500=4a;  +  700- 7^ ; 

whence,  3x=200 ; 

Thus  there  must  be  66|  lbs.  of  the  first  sort,  and  33^  lbs.  of  the  sec- 
ond sort. 

Ex.  9.  A  person  had  $5000,  part  of  which  he  lent  at  4 
per  cent.,  and  the  rest  at  5  per  cent. ;  the  whole  annual  in- 
terest received  was  $220 :  how  much  was  lent  at  4  per  cent.  ? 

Let  x=  the  number  of  dollars  lent  at  4  per  cent. ; 
then  5000— a;=  the  number  of  dollars  lent  at  5  per  cent. ; 


SIMPLE    EQUATIOXS.  149 

and  T7m~  ^^^^  aimual  interest  from  the  former; 

5(5000 x) 

and  '   —  the  annual  interest  from  tli£  latter; 

4x       5(5000 -.'?)     ^^^ 
therefore,  ___  +  1-__J=220; 

whence,  4x  +  5(5000-.t)=22000  ; 

that  is,  47?-f  25000-52=22000 ; 

/.   -a;=-8000,  or  2=3000. 
Thus  $3000  was  lent  at  4  per  cent. 

Ex.  10.  Divide  42  into  4  parts  which  shall  be  4  consecu- 
tive numbers. 

Let  X  be  one  part ; 
then  ic-{-l^  ^+2,  x-{-S,  are  tlie  other  parts ; 

and  X  +  (2+l)+(2+2)+(2+3)=42,  by  the  question  ; 

combining,  42j-|-6=42, 

or,  4^=36 ; 

.-.  2=9,  and  2+1=10,  2  +  2=11,  2+8=12 ; 
.*.  9,  10,  11,  12,  are  the  required  parts. 

181.  The  great  difficulty  which  the  beginner  finds  m  solv- 
ing these  problems  is  in  translating  the  statements  of  the 
enunciation  into  algebraical  language.  In  this,  practice  alone 
can  give  readiness  and  accuracy.  The  teacher  will  find  it 
advantageous  to  train  the  student  orally  in  such  transla- 
tions, by  means  of  examples  like  those  given  in  Art.  40. 

182.  The  student  should  always  read  carefully  and  con- 
sider well  the  meaning  of  the  question  proposed ;  and  in  or- 
der to  avoid  error,  he  should  observe  that  x  represents  an 
unJcno2vn  number  of  dollars,  poimds,  feet,  miles,  liours,  or  in 
general,  an  iinhnoivn  nwnher  of  tilings  or  units,  and  both 
the  land  and  denomination  of  the  units  of  x  should  be  dis- 
tinctly noticed  in  the  statement. 

What  caution  is  given  to  the  student  in  Art.  181? 


150  ELEMEJ^TARY    ALGEBRA. 

183.  Many  of  the  problems  given  below  may  be  solved 
readily  by  Arithmetic,  but  the  student  will  soon  perceive 
the  superiority  of  the  method  of  solution  by  Algebra,  in 
power  and  generality  and  easy  application. 

Examples — 36. 

1.  What  number  is  that  which  exceeds  its  sixth  part 
by  10? 

2.  What  number  is  that,  to  which  if  7  be  added,  twice 
the  sum  will  be  equal  to  32  ? 

3.  Find  a  number,  such  that  its  half,  third,  and  fourth 
parts  shall  be  together  greater  than  its  fifth  part  by  106. 

4.  A  bookseller  sold  10  books  at  a  certain  price,  and  after- 
ward 15  more  at  the  same  rate,  and  at  the  latter  time  re- 
ceived $8.75  more  than  at  the  former :  what  was  the  price 
per  book  ? 

5.  What  two  numbers  are  those,  whose  sum  is  48  and  dif- 
ference 22  ? 

6.  At  an  election  where  979  votes  were  given,  the  success- 
ful candidate  had  a  majority  of  47 :  what  were  the  num- 
bers for  each  ? 

7.  A  spent  62J-  cents  in  oranges,  and  says,  that  3  of  them 
cost  as  much  under  25  cts.,  as  9  of  them  cost  over  25  cts.  -■ 
how  many  did  he  buy  ? 

8.  The  sum  of  the  ages  of  two  brothers  is  49,  and  one  of 
them  is  13  years  older  than  the  other :  find  their  ages. 

9.  Find  a  number  such  that  if  increased  by  10  it  will  be- 
come five  times  as  great  as  the  third  part  of  the  original 
number. 

10.  Divide  150  into  two  parts,  so  that  one  of  them  shall  be 
two-thirds  of  the  other. 


SIMPLE    EQUATIONS.  151 

11.  A  child  is  born  in  November,  and  on  the  tenth  day  of 
December  he  is  as  many  days  old  as  the  month  was  on  the 
day  of  his  birth :  when  was  he  born  ? 

12.  There  is  a  number  such  that,  if  8  be  added  to  its 
double,  the  sum  will  be  five  times  its  half.     Find  it. 

13.  Divide  87  into  three  parts,  such  that  the  first  may  ex- 
ceed the  second  by  7,  and  the  third  by  17. 

14.  Find  a  number  such  that,  if  10  be  taken  from  its 
double,  and  20  from  the  double  of  the  remainder,  there  may 
be  40  left. 

15.  A  market-woman  being  asked  how  many  eggs  she  had, 
replied :  If  I  had  as  many  more,  half  as  many  more,  and  one 
egg  and  a  half,  I  should  have  104  eggs :  how  many  had  she  ? 

16.  A  is  twice  as  old  as  B ;  twenty-two  years  ago  he  was 
three  times  as  old.     Required  ^'s  present  age. 

17.  Divide  $64  among  three  persons,  so  that  the  first  may 
have  three  times  as  much  as  the  second,  and  the  third,  one- 
third  as  much  as  the  first  and  second  together. 

18.  A  workman  is  engaged  for  28  days  at  62^  cts.  a  day, 
but  is  to  pay  25  cts.  a  day,  instead  of  receiving  anything,  on 
all  days  upon  which  he  is  idle.  He  receives  altogether 
$13.12.j :  for  how  many  idle  days  did  he  pay  ? 

19.  A  person  buys  4  horses,  for  the  second  of  which  he 
gives  £12  more  than  for  the  first,  for  the  third  £6  more  than 
for  the  second,  and  for  the  fourth  £2  more  than  for  the 
third.  The  sum  paid  for  all  was  £230.  How  much  did 
each  cost? 

20.  A  person  bought  20  yards  of  cloth  for  10  guineas,  for 
part  of  which  he  gave  lis.  6d.  a  yard,  and  for  the  rest  75.  6d. 
a  yard :  how  many  yards  of  each  did  he  buy  ? 


152  ELEMEXTARY    ALGEBllA. 

21.  Two  coaches  start  at  the  same  time  from  A  and  B,  a 
distance  of  200  miles,  travellirng  one  at  9^  miles  an  honr, 
the  other  at  9^:  where  will  they  meet,  and  in  what  time 
from  starting? 

22.  A  father  has  six  sons,  each  of  whom  is  four  years  older 
than  his  next  younger  brother ;  and  the  eldest  is  three  times 
as  old  as  the  youngest :  find  their  respective  ages. 

23.  ^  is  twice  as  old  as  B,  and  seven  years  ago  their  united 
ages  amounted  to  as  many  years  as  now  represent  the  age  of 
A:  find  the  ages  of  A  and  B, 

24.  Two  persons,  A  and  B,  are  travelling  together;  A  has 
£100  and  B  has  £48:  they  are  met  by  robbers,  who  take 
twice  as  much  from  A  as  from  B,  and  leave  to  A  three  times 
as  much  as  to  i? :  how  much  was  taken  from  each  ? 

25.  Find  two  consecutive  numbers,  such  that  one-half  and 
one-fifth  of  the  first,  taken  together,  shall  be  equal  to  one- 
third  and  one-fourth  of  the  second,  taken  together. 

26.  A  cistern  is  filled  in  20  minutes  by  3  pipes,  the  first  of 
w^hich  conveys  10  gallons  more,  and  the  second  5  gallons  less, 
than  the  third,  per  minute.  The  cistern  holds  820  gallons. 
How  much  flows  through  each  pipe  in  a  minute  ? 

27.  A  garrison  of  1000  men  was  victualled  for  30  days ; 
dfter  10  days  it  was  re-enforced,  and  then  the  provisions  were 
exhausted  in  5  days:  find  the  number  of  men  in  the  re- 
enforcement. 

28.  In  a  certain  weight  of  gunpowder,  the  saltpetre  com- 
posed 6  lbs.  more  than  a  half  of  the  weight,  the  sulphur  5  lbs. 
less  than  one-third,  and  the  charcoal  3  lbs.  less  than  one- 
fourth  :  how  many  pounds  were  there  of  each  of  the  three 
ingredients  ? 

29.  A  general,  after  having  lost  a  battle,  found  that  he 
had  left,  fit  for  action,  3600  men  more  than  half  of  his  army; 
600  men  more  than  one-eighth  of  his  army  were  wounded ; 


SIMPLE    EQUATIONS.  153 

and  the  remainder,  forming  one-fiftli  of  liis  army,  were  slain, 
taken  prisoners,  or  missing:  what  was  the  number  of  the 
army  ? 

30.  A  tradesman  starts  v/ith  a  certain  sum  of  money :  at 
the  end  of  the  first  year  he  had  doubled  his  original  stock, 
all  but  £100;  also,  at  the  end  of  the  second  year  he  ha\ 
doubled  the  stock  at  the  beginning  of  that  year,  all  but 
£100;  also  in  like  manner  at  the  end  of  the  third  year;  -and 
at  the  end  of  the  third  year  he  found  himself  three  times  as 
rich  as  at  first :  what  Avas  his  original  stock  ? 


XXV.    Problems — continued. 

184.  We  shall  now^  give  some  examples  rather  more  diffi- 
cult than  the  examples  of  the  preceding  chapter. 

Ex.  1.  Find  a  number,  such  that  if  |  of  it  be  subtracted 
from  20,  and  y^  of  the  remainder  from  |  of  the  original 
number,  12  times  the  second  remainder  shall  be  half  the 
original  number. 

Let  X  =  the  number ; 

.-.  20— |.T=lst  remainder,  and  l-a'—-f^{20—^x)  =  2d  remainder; 

.*.  12[J:r-y^Y(20— |.?0]=4.r,  by  the  question;  whence  ^  =  24. 

Ex.  2.  A  certain  number  consists  of  two  digits  whose 
difference  is  3 ;  and,  if  the  digits  be  inverted,  the  number  so 
formed  will  be  ^  of  ^^^^  former :  find  the  original  number. 

Let  a;=lesser  digit,  and  .*. ^+3==ihe  greater:  then,  since  the  value  of 
a  number  of  two  digits = ten  times  the  lirst  digit+the  second  digit, 
(thus  67=10x0+7),  the  number  in  question=10  ( a'+a )  H-^ ;  similarly, 
the  number  formed  by  the  same  digits  inverted=10.2'+(.'c+3);  hence, 
by  question,  10c+(a?+3)=f[10(;c+3)+.i'],  whence  ii=3,  ic-\-'S=6,  and 
the  number  required  is  63. 

Ex.  3.  A  can  do  a  piece  of  work  in  10  days;  but,  after  he 
has  been  upon  it  4  days,  B  is  sent  to  help  him,  and  they 


154:  ELEMENT  AH  Y    ALGEBRA. 

finish  it  together  in  2  days.     In  what  time  would  B  have 
done  the  whole  ? 

Let  a'^u  umber  of  days  B  would  have  taken,  and  IT  denote  the  work : 

W  W 
.-.  — ,  — ,  are  the  portions  of  the  work  which  A,  B  would  do  in  one 
10    X 

4Tr 
day ;  hence  in  4  days,  A  does  — -r-,  and  in  2  days,  A  and  B  together 

^"  10+^=  •••  TO+lO +-^-^^'  whence  .==0. 

It  is  plain  that,  in  the  above,  we  might  have  omitted  W  altogether, 
or  taken  unity  to  represent  the  work,  as  follows : 

J.,  5  do  — ,  —  of  the  work,  respectively,  in  one  day,  and  therefore, 

4      2      2 
reasonino^  lust  as  before,  -7:+-^H —  =  the  whole  w^ork  =  1. 

[In  all  such  questions,  the  student  should  notice  that,  if  a  person 
does  — ths  of  any  work  in  one  day,  he  will  do  — th  of  it  in  — th  of  a  day, 

and  therefore  the  whole  work  in  —  days. 

Thus,  if  he  does  -|  in  one  day,  he  will  do  ^  in  -J-  of  a  day,  and  there- 
fore the  whole,  or  -^ ,  in  3^=2^  days.] 

Ex.  4.  A  alone  can  perform  a  piece  of  work  in  9  days,  and 
B  alone  can  perform  it  in  12  days :  in  what  time  will  they 
perform  it  if  they  work  together  ? 

Let  X  denote  the  required  number  of  days.    In  one  day  A  can  perform 

1  X 

— th  of  the  work  ;  therefore,  in  x  days  he  can  perform  ^ths  of  the  work. 

In  one  day  B  can  perform  r^  of  the  work ;  therefore,  in  x  days  he  can 

perform  T^ths  of  the  work.    And  since  in  x  days  A  and  B  together 

perfoi-m  the  whole  work,  the  sum  of  the  fractions  of  the  work  must  be 
equal  to  unity ;  that  is, 

9^12 

Multiplying  by  36.  4^'+3:i;=36, 

that  is,  7a?=36 ; 

36 
therefore,  if=— =5-JJ- 


SIMPLE   EQUATIOIs^S.  155 

Ex.  5.  A  cistern  could  be  filled  with  water  by  means  of 
one  pipe  alone  in  6  hours,  and  by  means  of  another  pipe  alone 
in  8  hours ;  and  it  could  be  emptied  by  a  tap  in  12  hours,  if 
the  two  pipes  were  closed :  in  what  time  will  the  cistern  be 
filled  if  the  pipes  and  the  tap  are  all  open  ? 

Let  X  denote  the  required  number  of  hours.     In  one  hour  the  first 

1  X 

pipe  fills  — th  of  the  cistern;  therefore,  in  x  hours  it  fills  Trths  of  the 

6  6 

cistern.     In  one  hour  the  second  i3ipe  fills  — th  of  the  cistern ;  therefore, 

o 

X  1 

in  X  hours  it  fills  ^ths  of  the  cistern.     In  one  hour  the  tap  empties  — rth 

X 

of  the  cistern ;  therefore,  in  x  hours  it  empties  T^ths  of  the  cistern.  And 
since  in  x  hours  the  tohole  cistern  is  filled,  we  haye 

XX        ^  _-, 

Multiplying  by  24,  4x  +  dx—2x=24:, 

that  is,  5x=24 ; 

therefore,  a*==^==4|-. 

185.  It  is  sometimes  convenient  to  take  x  to  represent  not 
the  quantity  which  is  actually  demanded  in  the  question, 
but  some  other  unknown  quantity  on  which  if  depends. 
This  we  will  illustrate  by  some  examples.  But  experience  is 
the  only  guide  to  the  best  selection  of  the  unknown  quantity. 

Ex.  6.  A  colonel,  on  attempting  to  draw  up  his  regiment 
in  the  form  of  a  solid  square,  finds  that  he  has  31  men  over, 
and  that  he  would  require  24  men  more  in  his  regiment  in 
order  to  increase  the  side  of  the  square  by  one  man :  how 
many  men  were  there  in  the  regiment  ? 

Let  X  denote  the  number  of  men  in  the  side  of  the  first  square ;  then 
the  number  of  men  in  the  square  is  .^''^,  and  the  number  of  men  in  the 
regiment  is  x"^  +  31.  If  there  were  x-hl  men  in  a  side  of  the  square,  the 
number  of  men  in  the  square  would  be  (x  +  lf ;  thus  the  number  of 
men  in  the  regiment  is  (:r'  +  l)'^— 24. 


156  ELEMEXTAEY    ALGEBRA. 


Therefore, 

(^+1)^- 

-24=. 

i''  +  31; 

that  is, 

x'  +  2x  +  l- 

-24- 

,v''  +  Sl. 

From  these 

two 

equal  expressions  we  can  remove  x"^, 

which 

occurs 

in  both ;  thus, 

2^  +  1 

-24= 

=31; 

therefore. 

2.r=31-l 

+  24= 

=54; 

or, 

a: 

54_ 

=27. 

Hence  the  number  of  men  in  the  regiment  is  (27f +  31;  that  is, 
729  +  31;  that  is,  760. 

Ex.  7.  A  starts  from  a  certain  place  and  travels  at  the  :ate 
of  21  miles  in  5  hours ;  B  starts  from  the  same  place  8  hoars 
after  A,  and  travels  in  the  same  direction  at  the  rate  of  15 
miles  in  3  hours :  how  far  will  A  travel  before  he  is  overtaken 
by  5? 

Let  a^=the  number  of  hours  which  A  travels  before  he  is  overtaken ; 
then  ic— 8=the  number  of  hours  B  travels. 

21 
Now  since  A  travels  21  miles  in  5  hours,  (i.  e.)  ~  of  a  mile  in  one 

o 

hour,  therefore, 

~— =  the  number  of  miles  which  A  travels  in  x  hours ; 
5 

15 

and  similarly,  —(^—8)=  number  of  miles  which  B  travels  In  x  horn's 

o 

Therefor^'  ^  (aj-8)=^ ; 

o  O 

25(«-8)=21a?; 

25iz;-21^=200 ; 

a— 50. 

21 V    21 
Therefore,  -—=— X  50=210 ;  so  that  A  travelled  210  miles  before 
5       5 

lie  was  overtaken. 

186.  The  principles  of  proportion,  as  taught  in  Arithmetic, 
are  often  used  to  form  the  equations. 

Ex.  8.   It  is  required  to  divide  the  number  60  into  three 
•   parts,  such  that  they  may  be  to  each  other  in  the  proportion 
of  the  numbers  3,  4,  and  5. 


SIMPLE    KOUATIOXS.  15? 

Let  the  number  x  denote  the  first  part.     Then,  since 

1st  part  :  2d  part  : :  8  :  4,  therefore  — -  is  the  second  part; 

o 

and  since  1st  part  :  3d  part  : :  3  :  5,  tlierefore  —  is  the  third  part 

o 

Therefore,  the  sum  of  the  parts 

x  +  -x-\--x—m', 

3a?  +  4aj  +  5aj=180;  , 
12.r=180 ; 

a:'=15,  the  first  part. 

4  5 

Hence  the  2d  part  is  —  Xl5,  or  20,  and  the  3d  part  is  ^Xl5,  or  25. 

o  o 

The  preceding  mode  of  solution,  and  many  similar  solu- 
tions involving  proportions,  may  be  shortened  after  the 
following  manner : 

Let  3.^  denote  the  first  part ;  then  the  second  part  must  be  4a;,  and 
the  third  part  must  be  5.r. 

Therefore,  ^x  +  ^x  +  ^i  =80 ; 

12,  =60;         a  =5. 

. .  3x5=15,  the  first  part;  4x5=20,  the  second  part;  5X5=25,  the 
third  part. 

Ex.  9.  There  are  two  bars  of  metal,  the  first  containing  14 
oz.  of  silver  and  6  of  tin,  the  second  containing  8  of  silver 
and  12  of  tin :  how  much  must  be  taken  from  each  to  form 
a  bar  of  20  oz.,  containing  equal  weights  of  silver  and  tin  ? 

Let  a:=number  of  oz.,  to  be  taken  from  first  bar,  '^0  —  x  from  second; 
now  ^0"  of  ^^^  fi^st  bar,  and  therefore  of  every  oz.  of  it,  is  silver ; 

and,  similarly,  -^  of  every  oz.  of  the  second  bar  is  silver ; 
Hud  there  are  to  be,  altogether,  10  oz.  of  silver  in  the  compound ; 

.'.  ii^'  +  ^o  (3O-.70=lO,  whence  .r=r)f,  and  20-.?=13i. 

Ex.  10.  Find  the  time  between  two  and  three  o'clock,  when 
the  minute-hand  of  a  watch  is  exactly  over  the  hour-hand. 
Find,  also,  the  time  between  2  and  3  o'clock,  when  the  hands 
are  exactly  opposite  each  other. 


158  ELEMEXTAilY     ALGEBEA. 

1st  Ciise. — Let  x  denote  the  required  number  of  minutes  after  2  o'clock. 
In  X  minutes  the  minute-hand  moves  over  x  divisions  of  the  watch-face ; 
and  as  the  long  hand  moves  12  times  as  fiist  as  the  short  hand,  the  latter 

"will  move  over  —  divisions  in  x  minutes.     At  2  o'clock  the  short  hand 

is  10  divisions  in  advance  of  the  long  hand ;  so  that  in  the  x  minutes 
the  long  hand  must  pass  over  10  more  divisions  than  the  short  hand. 

Therefore,  ^=T^  + 1^  5 

12^=^  +  120; 
110^=120; 

aj=-— 7-  =:10yt  minutes. 

Or  more  briefly,  thus :    The  minute-hand  in  every  minute  gains  — 

of  one  minute-division  on  the  hour-hand.    Hence,  in  x  minutes,  it  gains 

-r^r- divisions.    Therefore, -3-^=10.    :.  x=———10\f. 
1/^  \Z  11 

.  2d  Case. — Here  the  minute-hand  must  not  only  overtake  the  hour- 
hand,  but  advance  so  as  to  leave  it  30  minute-divisions  behind.  Then 
let  (C=  required  number  of  minutes  after  2  o'clock  ;  then  the  gain  of  the 

llx 
minute-hand,  or  — ^  =10  -\-  30. 

llx 
:.  -—-=40,  lla?=480 ;  x=A:^-^j  minutes.     Hence  the  hands  are  in  the 

required  position  at  43x\  minutes  past  2  o'clock. 

Ex.  11.  A  hare  takes  four  leaps  to  a  greyhound's  three, 
but  two  of  the  greyhound's  leaps  are  equivalent  to  three  of 
the  hare's ;  the  hare  has  a  start  of  50  leaps :  how  many  leaps 
must  the  greyhound  take  to  catch  the  hare  ? 

Suppose  that  ^x  denote  the  number  of  leaps  taken  by  the  greyhound ; 
then  Ax  will  denote  the  number  of  leaps  taken  by  the  hare  in  the  same 
time.  Let  a  denote  the  number  of  inches  in  one  leap  of  the  hare ;  then 
3rx  denotes  the  number  of  inches  in  three  leaps  of  the  hare,  and  there- 
fore also  the  number  of  inches  in  two  leaps  of  the  greyhound :  therefore 

3<2 

-—  denotes  the  number  of  inches  in  one  leap  of  the  greyhound.    Then 

^x  leaps  of  the  greyhound  will  contain  ^xX~  inches.    And  50-l-4iB 

leaps  of  the  hare  will  contain  (50-f-4.T)  a  inches;  therefore, 

^xa     _.      .  . 
— -=(oO  -I-  4a;)  a. 


SIMPLE   EQUATIONS.  159 

Dividing  by  a,  i7=50  4-  4c ; 

therefore,  9.^=100  +  8^; 

or,  .  a.'=100. 

Thus  the  greyhound  must  take  300  leaps. 

Here  an  auxiliary  symbol  a  has  been  introduced  to  enable  us  to  form 
the  equation  more  easily.  Being  in  every  term,  it  is  removed  by  di- 
vision Avhen  the  equation  is  formed. 

Ex.  12.  If  tlie  specific  gravity  of  pure  milk  be  1*03,  and 
a  certain  mixture  of  milk  and  water  be  found  (by  means  of 
an  instrument  for  the  purpose)  to  be  of  specific  gravity 
1*02625,  how  much  water  has  been  added  ? 

[Definition.— ^By  the  specific  gramty  of  a  substance  is  meant  the 
number  of  times  which  its  weight  is  of  an  equal  bulk  of  water.  Thus 
the  specific  gramty  of  silver  is  10"5,  or  10 j,  which  means  that  any 
quantity  of  silver  is  lOJ  times  the  weight  of  the  same  hulk  of  water. 
The  specific  gravity  of  milk  being  1.03,  signifies  that  milk  is  lyl-g- 
times  as  heavy  as  water ;  and  so  on.] 

Let  1  quart  of  water  be  added  to  x  quarts  of  pure  milk  to  form  the 
mixture;  then, 
Since  the  w^  eight  of  x  quarts  of  pure  milk 

=1 .03  times  the  weight  of  x  quarts  of  water, 

— 1.03X  X  X  weight  of  1  quart  of  water; 

.-.  whole  weight  of  water  and  milk 

=(1  +  1.03.r)X  weight  of  1  quart  of  water. 

But  there  are  1+x  quarts  of  the  mixture  whose  specific  gravity  is 
1.02625;  .*.  the  Avhole  weight  of  this 

=1.02625(1 +cr)X  weight  of  1  quart  of  water; 

.-.  l  +  1.03.T=1.02625(l  +  ir). 

Therefore,  (1. 03 -1.02625>r=l. 02625-1 ; 

that  is,  .00375.t;=.02625  ; 

_. 02625 
•*•  "^"".00375 

Hence,  1  quart  of  water  has  been  added  to  7  quarts  of  milk ;  (i.  e.) 
o7i€-eigWi  of  the  mixture  is  water. 


160  ELEMENTARY    ALGEBRA. 


Examples — 37. 


1.  Out  of  a  cask  of  wine  of  which  a  fifteenth  part  had 
leaked  away,  12  gallons  were  drawn,  and  then  it  was  two- 
thirds  full :  how  much  did  it  hold  ? 

2.  In  a  garrison  of  2744  men  there  are  two  cavalry  sol- 
diers to  twenty-five  infantry,  and  half  as  many  artillery  as 
cavalry  :  find  the  number  of  each. 

3.  The  first  digit  of  a  certain  number  exceeds  the  second 
by  4,  and  when  the  number  is  divided  by  the  sum  of  the 
digits,  the  quotient  is  7  :  find  it. 

4.  The  length  of  a  floor  exceeds  the  breadth  by  4  feet;  if 
each  had  been  increased  by  a  foot,  there  would  have  been  27 
more  square  feet  in  it :  find  its  original  dimensions. 

5.  In  a  mixture  of  copper,  lead,  and  tin,  the  copper  was 
5  lbs.  less  than  half  the  whole  quantity,  and  the  lead  and 
tin  each  5  lbs.  more  than  a  third  of  the  remainder :  find  the 
respective  quantities. 

6.  A  horse  was  sold  at  a  loss,  for  $210;  but  if  it  had 
been  sold  for  $262.50,  the  gain  would  have  been  three- 
fourths  of  the  former  loss :  find  its  real  value. 

7.  A  can  do  a  piece  of  work  in  10  days,  which  B  can  do 
in  eight ;  after  A  has  been  at  work  upon  it  3  days,  B  comes 
to  help  him  :  in  what  time  will  they  finish  it  ? 

8.  There  is  a  number  of  two  digits  whose  difference  is  2, 
and,  if  it  be  diminished  by  half  as  much  again  as  the  sum 
of  the  digits,  it  will  give  a  number  expressed  by  the  digits 
inverted :  find  it. 

9.  A  number  of  troops  being  formed  into  a  solid  square, 
it  was  found  that  there  were  60  over;  but  when  formed  into 
a  column  with  five  men  more  in  front  than  before,  and  three 
less  in  depth,  there  was  just  one  man  wanting  to  com- 
plete it :  what  was  the  number  of  troops  ? 


SIMPLE    EQUATIONS.  161 

10.  A  person  has  travelled  altogether  3036  miles,  of  which 
he  has  gone  seven  miles  by  water  to  four  on  foot,  and  five  by 
Avater  to  two  on  horseback:  how  many  did  he  travel  each 
way  ? 

11.  A  mass  of  copper  and  tin  weighs  80  lbs.,  and  for  every 
7  lbs.  of  copper  there  are  3  lbs.  of  tin  :  how  much  copper 
must  be  added  to  the  mass,  that  there  may  be  4  lbs.  of  tin 
for  every  11  lbs.  of  copper? 

12.  A  does  I  of  a  piece  of  work  in  10  days,  when  B  comes 
to  help  him,  and  they  take  three  days  more  to  finish  it :  in 
what  time  would  they  have  done  the  whole,  each  separately, 
or  both  together  ? 

13.  A  and  B  were  employed  together  for  50  days,  each  at 
$1.20  a  day;  during  which  time  A,  by  spending  12  cents  a 
day  less  than  B,  had  saved  three  times  as  much  as  B,  and  2^ 
days'  pay  besides :  what  did  each  spend  per  day  ? 

14.  There  are  two  silver  cups,  and  one  cover  for  both  ;  the 
first  weighs  12  oz.,  and  with  the  cover  weighs  twice  as  much 
as  the  other  cup  without  it ;  but  the  second  with  the  cover 
weighs  a  third  as  much  again  as  the  first  without  it :  find 
the  weight  of  the  cover. 

15.  Find  a  number  of  three  digits,  each  greater  by  unity 
than  that  which  follows  it,  so  that  its  excess  above  one- 
fourth  of  the  number  formed  by  inverting  the  digits  shall  be 
36  times  the  sum  of  the  digits. 

16.  If  19  lbs.  of  gold  weigh  18  lbs.  in  water,  and  10  lbs. 
of  silver  weigh  9  lbs.  in  water,  find  the  quantity  of  gold  and 
silver  respectively  in  a  mass  of  gold  and  silver  weighing  106 
lbs.  in  air  and  99  lbs.  in  water. 

VI,  A  and  B  can  reap  a  field  together  in  12  hours,  A  and 
(7  in  16  hours,  and  A  by  himself  in  20  hours  :  in  what  time 
could,  1st,  B  and  C  together,  and,  2d,  A,  B,  and  G  to- 
gether, reajf  it? 


162  ELEMEXTARY    ALGEBRA. 

18.  Find  two  numbers  whose  difference  is  4,  and  the  dif- 
ference of  their  squares  112. 

19.  Divide  the  number  88  into  four  parts,  such  that  the 
first  increased  by  2,  the  second  diminished  by  3,  the  third 
multiplied  by  4,  and  the  fourth  divided  by  5,  may  all  be 
equal. 

20.  Three  persons  whose  powers  for  w^ork  are  as  the  num- 
bers 3,  4,  6,  can  together  complete  a  piece  of  work  in  60 
days :  in  what  time  could  each  alone  complete  the  work  ? 

21.  A  and  B  are  at  present  of  the  same  age;  if  ^'s  age  be 
increased  by  36  years,  and  ^^s  by  52  years,  their  ages  will  be 
as  3  to  4 :  w^hat  is  the  present  age  of  each  ? 

22.  Divide  100  into  two  parts,  such  that  the  square  of  their 
difference  may  exceed  the  square  of  twice  the  less  part  by 
2000. 

23.  A  cistern  has  two  supply-pipes  which  will  singly  fill 
it  in  4|-  hours,  and  6  hours,  respectively ;  and  it  has  also  a 
leak,  by  which  it  would  be  emptied  in  5  hours:  in  how  many 
hours  will  it  be  filled  when  all  are  working  together  ? 

24.  A  market-woman  bought  a  certain  number  of  eggs  at 
the  rate  of  5  for  2  cents ;  she  sold  half  of  them  at  2  for  a 
.cent,  and  half  of  them  at  3  for  a  cent,  and  gained  4  cents  by 
so  doing :  what  was  the  number  of  eggs  ? 

25.  A  and  B  shoot  by  turns  at  a  target ;  A  puts  7  bullets 
out  of  12  into  the  bull's-eye,  and  B  puts  in  9  out  of  12 ;  be- 
tween them  they  put  in  32  bullets :  how  many  shots  did 
each  fire  ? 

26.  Two  casks,  A  and  B,  contain  mixtures  of  wine  and 
water ;  in  A  the  quantity  of  wine  is  to  the  quantity  of  water 
as  4  to  3 ;  in  B  the  like  proportion  is  that  of  2  to  3.  If  A 
contains  84  gallons,  what  must  B  contain,  so  that  when  the 
two  are  put  together  the  new  mixture  may  be  half  wine  aud 
half  water?  "^ 


SIMPLE   EQUATION'S.  '        163 

27.  How  many  minutes  does  it  want  to  4  o'clock,  if  three- 
quarters  of  an  hour  ago  it  was  twice  as  many  minutes  past 
two  o'clock? 

28.  What  is  the  time  after  6  o'clock  at  which  the  hands  of 
a  watch  are,  1st,  directly  o23posite,  and,  2d,  at  right  angles 
to  each  other  ? 

29.  It  is  between  11  and  12  o'clock,  and  it  is  observed  that 
the  number  of  minute-spaces  between  the  hands  is  two- 
thirds  of  what  it  was  ten  minutes  previously :  find  the  time. 

30.  The  national  debt  of  a  country  was  increased  by  one- 
fourth  in  a  time  of  war.  During  a  long  peace  which  fol- 
lowed, $125,000,000  was  paid  off,  and  at  the  end  of  that 
time  the  rate  of  interest  was  reduced  from  4|-  to  4  per  cent. 
It  was  then  found  that  tfle  amount  of  annual  interest  was 
the  same  as  before  the  war :  what  was  the  amount  of  debt 
before  the  war  ? 

31.  Find  three  numbers,  the  sum  of  which  is  70,  and  such 
that  the  second  divided  by  the  first  gives  2  for  quotient,  and 
1  for  remainder ;  and  the  third  divided  by  the  second  gives 
3  for  quotient,  and  3  for  remainder. 

32.  Shells  are  thrown  from  two  mortars  in  a  besieged  city ; 
the  first  mortar  has  thrown  36  shells  before  the  second  com- 
mences its  fire,  and  it  sends  8  shells  for  every  7  sent  by  the 
second ;  but  the  second  expends  as  much  powder  in  3  dis- 
charges as  the  first  does  in  4 :  how  many  shells  must  the  sec- 
ond mortar  throw  in  order  to  expend  the  same  amount  of 
powder  as  the  first  ? 

187.  We  shall  now  give  a  few  problems  in  which  the 
known  quantities  are  represented  by  the  first  letters  of  the 
alphabet,  instead  of  numbers. 

Examples — 38. 

1.  Find  a  number,  such  that  being  divided  successively  by 
m  and  n,  the  sum  of  the  quotients  shall  be  equal  to  a. 


T64  ELEMENTARY    ALGEBRA. 

2.  Divide  a  number  a  into  two  such  part3  tV^ai  the  quo- 
tient of  the  one  divided  by  m,  and  the  other  divided  by  n, 
may  be  equal  to  h. 

3.  Divide  a  number  a  into  two  parts  proportional  to  the 
numbers  m  and  n, 

4.  Divide  a  number  a  into  three  parts,  such  that  the  first 
may  be  to  the  second  as  m  is  to  n^  and*  the  second  to  the 
third  asj9  is  to  q. 

5.  Two  numbers,  a  and  I,  being  given,  what  number  must 
be  added  to  each  one  of  them  in  order  that  the  ratio  of  the 

'}7l 

two  sums  may  be  equal  to  —  ? 

6.  'Three  fountains  will  fill  a  ce]i;ain  reservoir,  when  each 
one  runs  alone,  in  the  times  a,  t,  and  c,  respectively.  In 
what  time  will  they  fill  it,  all  running  together  ? 

7.  Two  couriers,  whose  distance  apart  >  when  they  set  out 
was  d  miles,  travel  toward  each  other,  the  one  moving  at  the 
rate  of  h  miles  an  hour,  and  the  other  at  the  rate  of  c  miles 
an  hour.     In  what  time  after  starting  will  they  meet  ? 

8.  A  can  do  a  piece  of  work  in  h  days,  and  B  can  do  the 
same  work  in  c  days.  In  what  time  can  they  together  do  the 
work? 

XXYI.      SiMUIiTANEOUS  EQUATIONS  OF  THE  FlRST  DEGREE. 

188.  If  one  equation  contain  tivo  unknown  quantities, 
there  are  an  infinite  number  of  pairs  of  values  of  these  by 
which  it  may  be  satisfied. 

Thus  in  .t=10  — 2?/,  if  we  give  any  value  to  y,  we  shall  get 
a  corresponding  value  for  x,  by  which  pair  of  values  the 
equation  will  of  course  be  satisfied.     If,  for  example,  we  take 

When  one  equation  contains  two  unknown  quantities,  what  values  may  they 
have? 


SIMULTANEOUS   EQUATIONS.  165 

t/—l,  we  shall  get  :?;r=:10  — 2  =  8;  if  y^2,  x=z6;  if  ^=3, 
x=4cf  &c. 

One  equation  then,  containing  ttvo  unknown  quantities 
(or,  as  it  is  expressed,  "between  two  unknown  quantities"), 
admits  of  an  infinite  number  of  solutions ;  but  if  we  have 
as  many  different  equations  as  there  are  quantities,  the  num- 
ber of  solutions  will  be  limited ;  for  it  will  be  seen  that  they 
can  always  be  reduced  to  a  single  equation  containing  a  sin- 
gle unknown  quantity. 

Thus,  while  each  of  the  equations,  2:=  10— 2^,  4:X=:32  —  6y, 
separately  considered,  is  satisfied  by  an  infinite  number  of 
pairs  of  values  of  x  and  y,  we  shall  find  there  is  only  one 
value  of  X,  and  one  value  of  y,  which  will  satisfy  both  equa- 
tions; for,  multiplying  the  first  equation  by  3, 

dx=30  —  6y;  now  take  this  from 

the  second  equation  4:^=32  —  6^,  and  we  get  x=i2. 

Thus  2; =2  is  the  only  value  of  x  common  to  both  equa- 
tions. Put  this  value  of  ;c  in  either  of  the  two  given  equa- 
tions— for  example,  in  the  first;  and  we  obtain, 

2=^10-2^; 

/.  2y=S;        .-.  y=^. 

Thus,  x=2,  y=^4:,  are  the  only  pair  of  values  which  satisfy 
both  equations. 

189.  Equations  of  this  kind,  which  are  to  be  satisfied  by 
the  same  pair  or  pairs  of  values  of  x  and  y,  are  called  simul- 
taneous equations.  In  the  present  chapter  we  treat  of  si- 
multaneous equations  of  the  first  degree,  (i.  e.)  where  each 
unknown  quantity  occurs  only  in  the  first  power,  and  the 
product  of  the  unknown  quantities  does  not  occur.  If 
there  be  three  unknown  quantities  there  must  be  three  equa- 
tions, and  so  on. 

Simultaneous  EquaUons. 


166  ELEMEl^TARY     ALGEBRA. 

190.  These  simultaneous  equations  must  all  express  dijf  ev- 
ent relations  between  the  unknown  quantities. 

Thus,  if  we  had  the  equation  x=10—2y,  it  would  be  of 
no  use  to  join  with  it  the  equation  2iz;=20— 4y  (which  is  the 
double  of  the  former),  or  any  other  deriyed  like  this  from 
the  former. 

191.  There  are  generally  given  three  methods  for  solving 
simultaneous  equations  of  two  unknowns;  but  the  object 

•aimed  at  is  the  same  in  each,  viz.,  to  combine  the  two  equa- 
tions in  such  a  manner  as  to'  expel,  or  eliminate,  one  un- 
known from  the  result,  and  so  get  an  equation  of  07ie  un- 
known only. 

192.  First  Method. — Multiply  fhe  equations  by  the  least 
numbers  wliicli  will  maJce  the  coefficients  of  07ie  of  the  tin 
Icnoivn  quantities  the  same  in  both  resulting  equations ;  then 
adding  or  subtracting  the  two  equatio7is  thus  obtained,  accord 
mg  as  the  equal  terms  have  different  or  the  same  signs,  these 
terms  ivill  destroy  each  other,  and,  the  elimination  will  be  ef- 
fected. 

Ex.1.  ^x-vZy^     4  1(1) 

3a:-2^=-7)  (2) 

Multiply  (1)  by  3,       6a; +9^/=     12. 

Multiply  (2)  by  2,       62;-4y--14. 

Subtracting,  13?/=26  and /.  ^=2. 

Then  put  this  value  of  y  in  either  (1)  or  (2) ;  for  example, 
in  (1).    We  have  thus: 

2a;  +  6=4;     .-.  2a;==4-6=:-2;     .-.  a:=-l. 

Ex.  2.  8a;+7?/=100  )  (1) 

12a;- 5y=  88  )  (2) 

What  is  said  of  the  relations  expressed  by  simultaneous  equations  ?    Of  the  ohject 
aimed  at  by  each  of  the  three  methods  of  solving  them  ?    First  method  ? 


SIMULTANEOUS   EQUATI02s^S.  167 

We  might  multiply  (1)  by  12  and  (2)  by  8,  giving  thus : 

96x-4:0y=  704 
Subtracting,  124^=  496         .-.«/  =  4. 

•  But  the  process  is  more  simple  if  we  multiply  equation  (1) 
by  3  and  equation  (2)  by  2.     Thus : 

24:X-\-21y==300 
24.x-10tj=176 

3 1^=124;  .•.^=4;    and,    substituting 
m  (1),  8a;+28  =  100;  /.  Sx=z'72,  and  x=9. 

We  see  here  the  advantage  of  multiplying  by  the  leasi 
numbers  which  will  make  the  coefficients  the  same,  though 
we  may  multiply  by  any  nmnbers  which  will  effect  the  same 
object. 

It  is  sometimes  possible  to  multiply  one  of  the  given  equa- 
tions by  some  number  which  will  make  the  coefficient  of  x 
or  y  in  it  the  same  as  in  the  other  equation.  The  process  is 
in  this  case  much  shortened. 

Ex.3.  4.x -^y^U)  (1) 

4y+a;=16j  (2) 

Here  multiplying  (2)  by  4,   16^  +  42^=64; 

but    y+jtx=U;     (1) 
/.  subtracting,  16y         =30,  and  ,\y=2-^ 

and  (2)  .r=:16-4y  =  16-8=8. 

Ex.  4.  4:x-  y=  7)  (1) 

dx  +  4:y=29  )  (2) 

Here  3.T  +  4y=29, 

and,  multiplying  (1)  by  4,     16a;— 4^=28; 

.*.  adding,  19a;         —57,  and  ,\x=d; 

and  (1)     ^=4a;~7n=  12-7=5.. 


168  ELEMEKTARY    ALGEBRA. 

193.  Second  Method. — Express  one  of  the  unhnoiun  quan- 
fities  171  terms  of  the  other  by  means  of  one  of  the  equations^ 
and  put  this  expression  for  it  in  the  other  equation. 

Thus,  taking  the  example  1  in  the  preceding  article, 

2.T  +  3?/=:     4)(1)  • 

^x-^^-l)  (2) 

4—3^ 
From  {l),x^=z — ^r-^;  substituting  this  expression  in  (2),  we 

obtain 

whence  3(4— 3?/)— 4^=  — 14; 

that  is,  12-9«/-4^=-14;      /.  -133/^-26;      .-.  iy=2. 

4-3y     4-6 
,,x-     ^     _    ^    _     i. 

Ex.  5.     7^'  +  i(2y  +  4)  =  16  )    or  reducing,  35:^  +  2;/=  76  )    (1) 
3y-i(a:  +  2)=  8)  12^-  a;=:34j    (2) 

Here  from  (2),  i?;=:12^-34,  and  from  (1),  35(12i/-34)+2«/ 
=  76;  whence  y=^,  and  /.  ir=2. 

194.  Third  Method. — Express  the  same  unlcnoivn  quantity 
in  terms  of  the  other  in  both  equatio7is,  and  put  these  expres- 
sions equal 

Thus,  taking  again  Ex.  1,     2x  +  3y=     4  )  (1) 

3x-2y=-7)  (2), 

(1)  gives  y=—^;     (2)  gives  y=-^-; 

^,       ^  3a; +  7     4.—2X 

therefore,  — - — = — - — . 

Second  method  ?    Third  method  ? 


SIMULTAXEOUS    EQUATIOKS.  1G9 

Clearing  of  fractions, 

9x  +  21=zS-4.x.     .\13x=z-r6,     x=-^. 

Ex.6.  5.7;-i(5y  +  2)  =  32)    or  reducing,  20.-?;— 5^=130  )    (1) 
3?/+t(^  +  2)=  9)  9ij+  x=  25)    (2) 

Here  in  (1),        y=^{20x-ld0),  in  (2),  y=^{2o-x) ; 

.^|(20.^•-130)3^i(25-^),  whence  a;=7,  y=^2. 

The  first  meiliod  is  to  be  preferred  generally;  but  the  second 
may  be  used  with  advantage  whenever  either  x  or  y  has  a  co- 
efficient unity  in  one  of  the  equations. 

Note. — It  may  be  well  to  give,  here,  an  abbreviation  of  the  first 
method,  which  saves  much  trouble  when  the  coefficients  are  large. 
Two  examples  will  serve  to  illustrate  it. 


Ex.  54^-121^=15  )  (1) 

^ .  y  to  find  X  and  y. 


36a?-88?/=-12;) 
36a.'- 77^=    21  ;f 


(2) 

Subtracting,  1 8a?—  44y=  -  6 ; 

multiplying  by  2, 

from  2d  equation, 

subtracting,  11^=33 ; 

And  I8.r=44?/-G=132-6=126;    .'.  a'=7. 

Ex.  101aj-242/=G3  )  d) 

103..-28^=29f^?^^^^"^^^- 
Subtracting,  2x-  4?/= -34; 

multiplying  by  6,  12^'—  24?/=  -  204 ;  ) 

but  101a;- 24y=      63;  ) 
subtracting,  89a;  ==267 ; 

267    ^ 

And  4^=2aj  + 34=40 ;     .-.  y=10. 
Which  method  is  preferable  ? 

8 


170  £Lemp:ktar.y   algebra. 

195.  Ex.  7.     Solve  —+-=:=8l    (1) 
X      y  ^ 


^^-1?  =  3!    (2) 
X      y        J 

If  we  cleared  these  equations  of  fractions  they  would  con- 
tain X  y,  and  could  not  then  be  solved  by  the  methods  of  this 
chapter.  But  if  we  do  not  clear  them  of  fractions,  they  may 
be  solved  readily  by  the  methods  given.     Thus, 


multiplying  (1)  by  3, 

'-V?-*=24; 
X      y 

multiplying  (2)  by  2, 

X      y        ' 

adding, 

-=.30;  .-.90= 

X           ' 

.-.  (1)  gives 

o 

therefore,         —  =  8  —  4 = 

=4;  .-.8=4./;  . 

=2. 

Ex.  8.     Solve  a''x-\-Vy:=^c''  (1),  ax-^hy^c  (2). 

Multiplying  (2)  by  Z>,  and  subtracting  it  from  (1), 
(^x^-h^y—& 
abx-\-h''y  —  'bG 

G^x—abx—c^  —  'bc\ 
that  is,  a(a—h)x—c{C'—y)\ 

'      ~a(a-by 
substituting  this  value  of  x  in  (2), 

acic—V)     7 

~f iT  +  oy=c\ 


in 

therefore, 


SIMULTANEOUS 

EQUATIOJfS. 

hy-~ 

c{c- 

-b) 

c(a- 

-c) 

'  («- 

h)- 

a— 

-b 

:.  ij~ 

c{a—c) 

Or  this  value  of  y  might  be  found  in  the  same  way  as  that 
of  X  was  found. 


Examples — 39. 

Find  the  values  of  x  and  y  in  the  following  pairs  of 
equations : 

1.  3a:-4y=2,  Ix-^y^^t, 

2.  7^-5?/=24,        4.x-3y=ll. 

3.  3^^+2^=32,         20x-3y=:l, 

4.  llx-7y=37,        8^+%=41. 

5.  '7x-i-5y=^60,  13:r-lly=:10. 

6.  6:^-7^=:.  42,  7x-6y  =  75. 

9.    V+    2    =^'        2+V=°- 

4~3~'  3  +  6~ 

„      l-3a;    3?/— 1     „  dx  +  y 

11.     -nj^  +  -Y-=2'         ^^  +  .V=9- 

13.     3(3a;  +  3j/)  =  3(2a;-32/)+10,      4a;-3y=4(6y-2a;)4-3. 


172  ELEMENTARY    ALGEBRA. 

13.     x{ij^7)=y{x  +  l)l  14.     i.^+i7/  =  13) 

2a;4-20  =  3^+l      )  :^x  +  i2j=D) 

15.  1.^+^^=43) 

16.  3.^+%=2.4,        ,21x-My=z.03. 

17.  ,dx  +  A26tj=x-6,        3x-.6y^2S—,26i/. 

18.  ,0Sx-.21tj=.S3,        .12x-}-.'7y=3M. 

x      y  X      y 

20.  — +-|-=2,        Ix—ay—^. 

21.  ir  +  ^=:<2+Z>,         'bx-\-ay—2ab, 

22.  -  +  4=1'         -^  +  -^=1. 
ah  0      a 

23.  {a-]-c)x—'by—'bc,        x-\-y—a-\-h, 

a      0  ha 

25.  x-}-y=c,        ax—hy—c{a—h), 

26.  «  (^  +  ^)+Z>(^— ^)  =  1,         a  {x~y)+h  {x-{-y)  =  l. 

27.   ?i:i^+l(z:^=o,      ?±i/:i^+?i=lr^=o. 


196.  Simultaneous  equations  of  three  unknown  quantities 
are  solved  by  eliminating  one  of  the  unknown  quantities  by 
means  of  any  pair  of  the  equations,  and  then  the  same  un- 
known by  means  of  another  pair;  we  shall  then  hare  two 
equations  involving  only  two  unknown  quantities,  which  we 
may  solve  by  the  preceding  rules.  The  remaining  unknown 
is  found  by  substituting  the  values  obtained  for  the  other 
two  in  any  of  the  given  equations. 

Simultaneous  equations  of  three  or  more  unknown  quantities. 


SIMli/rAXEOUS    EQUATIONS.  173 

Similarly  for  simultaneous  equatiojis  of  more  than  three 
unknown  quantities. 


Ex.1. 

x-2y+  3^=2]   (1) 
2^-3^+     z=l  I  (2) 
dx-  y\  ?.z-^\   (3) 

From  (1) 

2.^--4y+  6^=4 

(^) 

"^x-Zy^     z=l 
•.       -  y+  5^=3     (4) 

Again,  from 

(1) 

dx—6y+  9z=6 

(3) 

3x-  y+  2z=d 
\       -6y+  7;2=-3     (5) 

but  from  (4) 

-6y  +  25z=15 

—18^=— 18,  and  z=l 

hence  (4), 

y=  6z-3=2 

and  (1), 

x=2+2y-3z=:2  +  4:-3=S, 

Ex.2. 

1+^-^=1   (1) 

t+±  +  l=2^    (2) 
X      y      z  ^  ^ 

I-l+i=14    (3) 
X      y      z  ^  ' 

Multiplying  (1)  by  2,     |  +  A^|  =  2 

5      4 

-  +  — ■. 
X      y      z 

^  +  -^=36         (4) 
Multiply  (1)  by  3,  |+|_|=3 


Adding  (3)  to  this,         ^  +  -+-=24: 


174 

ELEMENT AKY 

ALGEBKA. 

Add 

(3)  to  this, 

1 
X 

y    " 

=  14 

10 

X 

_2_ 

y" 

-.11 

(i 

NoTv 

'  multiply  (5)  by  4, 

40 

X 

8_ 

'y~ 

:68 

Add 

(4)  to  this. 

X      y 

:26 

47_ 
X  ~ 

--M- 

,-.47=! 

__47_1 
•''-94-2' 


Substitute  the  value  of  x  in  (5) ;  thus, 

20--=17;       /. -=20-17^:3;     /.  v=-|-. 

y  y  '      ^    z 

Substitute  the  values  of  x  and  y  in  (1) ;  thus, 
Ex.3.     Solve  -+^=1     (1) 


From  (1)  -+-|-=1 

subtract  (2)  — I —  —  1; 


a      c 


to  the  result, 
add  (3) 


y 

z 

=  0 

b 

c 

z 
c 

=1 

^=1         y-^ 


Sr^iULTAXEOUS    EQUATIONS.  IvO 

Substitute  this  value  in  (1) ;  thus, 

Substitute  the  same  yalue  in  (3) ;  thus, 

1       z  z      1  __c 

These  values  of  z  and  x  might  have  been  written  down  at 
once  from  the  symmetry  of  the  equations,  since  it  is  obvious 
that  the  values  of  x  and  z  will  be  of  the  same  form  as  that 
of  y,  only  interchanging  a  and  c,  respectively,  with  h. 

Examples — 40. 

1.  2a;-h3v  +  4^=20l  2.  5^  +  3^^:65^ 

3a:+4y  +  5;^=26  I  2y-   z=ll 

3:?;  + 5^  +  6^=31  J  3^;+  4;2=57^ 

3.  dx^-2y-  ^=201  4.  x^y-^z=b^ 

'2x-^dy-\-Qz='10  \  x-\-y=z-l 


X- 


y-\-Qz—Al\  x-3=y  +  zj 


5.     x  +  2y=7  ^  6.  xy=x  +  y 

y-{-2z=2  I  xz=2{x  +  z) 

dx+2y=z—i  J  y^=^(y+^). 

7.  2(2;-^)  =  3^-2   1  8.  ix  +  iy=12-iz 

x  +  l  =  ^y+z)  I  iy+iz=  S+ix 

2.r+3^=4(l-^)J  ix  +  iz=10 

9.  y  +  iz=:ix  +  5 
i(^-l)-i(y-^)-TV(^  +  3) 
t.-i(2t/-5)=:lj-^^. 


176  ELEMEXTARY    ALGEBRA. 

10. =  -,  -  +  -  =  3|,  -  +  -  =  -. 

X      y      h  y      z        ^  X      y      z 

11.  y-\-z=a,         z-\-x=h,        x-\-y=c, 

12.  x-\-y-\-z—a-^h-\-Cy  x  +  a—y^h—z-\-c. 

13.  y^z—x=a,         z^x—y—l^         x  +  y—z—c. 

^ ,     X      y      z     ^  X      y      z     ^  x      y      z      . 

a      o      €  a      c      0  0      a      c 

KoTE. — In  Ex.  6,  divide  the  equations  by  xy,  xz,,  and  yz,  respectively, 
and  they  will  then  be  of  the  form  of  those  given  in  Ex.  10. 


XXVII.    Problems  solved  by  Simultaneous  Equations 
OF  THE  First  Degree. 

197.  Prob.  1.  There  is  a  certain  fraction  which  becomes 
equal  to  -J-  when  both  numerator  and  denominator  are  dimin- 
ished by  1 ;  but,  if  2  be  taken  from  the  numerator  and  added 
to  the  denominator,  the  fraction  becomes  equal  to  ^ :  find  it. 

Let  X  denote  the  numerator,  and  y  the  denominator  of  the  required 
fraction ;  then  the  conditions  of  the  problem  give, 

f-i~Y'        y  +  2~Y' 

Clear  the  equations  of  fractions, — transpose,  and  reduce.  We  obtani 
thus, 

2x-y=l    (1) 

dx-y=S    (2) 

Subtracting(l)from(2)  we  getir=7;  .'.  (1)  gives  14— 2/= 1;  .*.  2^=1«>; 
therefore,  the  required  fraction  is  J^. 

Prob.  2.  There  is  a  certain  number  composed  of  two  fig- 
ures or  digits,  which  is  equal  to  four  times  the  sum  of  its 
digits ;  and  if  the  digits  exchange  places,  the  number  thus 
formed  is  less  by  12  than  twice  the  former  number:  what  is 
the  number? 


SIMULTANEOUS    EQUATION'S.  177 

Let  X  be  the  digit  in  the  tens^  place, 
y ". . . .  ^m^ts^ ; 

then  lO-r+y  is  the  number  (just  as  23  =  10x2-f3),   /.  by  the  condi- 
tions of  the  question, 

10.zj  +  2/=4(.i'  1-2/), 

that  is,  =4:X-\Ay\ 

transposing,  10x  —  4:X=A:y—y\ 

uniting,  ^x=^y\ 

or,  2x=y    (1). 

Again,  if  the  digits  be  reversed,  10y-{-x  will  be  the  number ;  /.  by 
the  question, 

102/  +  ^=2(103:-l-^)-12; 
that  is,  =20.2^+2^-12; 

transposing,  l^x—Sy=VZ\ 

or,       [•••  2/=2^',  (1)],  mc-\Qx=12',  (2) 
uniting,  3.r=12; 

.'.  ir=4;  and  ^=2.2?= 8. 
. .  the  number  requu'ed  is  48. 

pROB.  3.  A  railway  train  after  travelling  an  hour  is  de- 
tained 24  minutes,  after  which  it  proceeds  at  six-fifths  of  its 
former  rate,  and  arrives  15  minutes  late.  If  the  detention 
had  taken  place  5  miles  further  on,  the  train  would  have  ar- 
rived 2  minutes  later  than  it  did.  Find  the  original  rate  of 
the  train,  and  the  distance  travelled. 

Let  5.^'  denote  the  number  of  miles  per  hour  at  which  the  train  origi- 
nally travelled,  and  let  y  denote  the  number  of  miles  in  the  whole  dis- 
tance travelled.  Then  y—t)X  will  denote  the  number  of  miles  w^hicli 
remain  to  be  travelled  after  the  detention.    At  the  original  rate  of  the 

train  this  distance  would  be  travelled  in  -~ —  hours :  at  the  increased 
rate  it  will  be  travelled  in  ^ —  hours.     Since  the  train  is  detained  24 

K)X 

minutes,  and  yet  is  only   15   minutes   late    at  its   arrival,  it   follows 

8* 


1T8  ELEMKJSTAKY    ALGEBiiA. 

that  the  remainder  of  the  journey  is  performed  in  9  minutes  less  than 

it  would  have  been  if  the  rate  had  not  been  increased ;  and  9  minutes 

9 
is  7^  of  an  hour ;  therefore, 
00 

y—6x_y-6x 9^ 

'~Q^~~5x~     60    ^  ^* 

If  the  detention  bad  taken  place  5  miles  further  on,  there  would 
have  been  y—6x—6  miles  left  to  be  travelled.  Thus  we  shall  find 
that 

y—6x-5_y-5x-5     7 

Qx      ~      6x  60    ^  ^' 

Subtracting  (2)  from  (1), 

Qx~5x    60' 

therefore,  50=60-2aj; 

whence,  2a?=10;  and  x=5. 

Substitute  this  value  of  a?  in  (1),  and  it  will  be  found  by  solving  the 
equation  that  y=4!7^. 

Prob.  4.  A  and  B  can  together  do  a  piece  of  work  in  a 
days;  A  and  0  can  together  do  it  in  b  days;  B  and  C  can 
together  perform  it  in  o  days:  find  the  number  of  days  in 
which  each  alone  could  perform  the  work. 

Let  X  denote  the  number  of  days  in  which  A  alone  could  perform 
it,  y  the  number  of  days  in  which  B  alone  could  perforai  it,  z  the  num- 
ber of  days  in  which  C  alone  could  perform  it. 

Then  we  have, 

X      z      b     ^  ^'  * 

1.1=1    (3). 

y      z      c 

Bubtractmg  (2)  fi-om  (1)  we  obtain, 

1_1=1_1    (4) 
y      z      a      b      ^  ' 


SIMULTAIS^EOUS    EOTATIOXS.  170 


1     1  _  1  ^ 

y      z      c  ' 

2_1       1 . 1 

y      a      b       c 

bc  +  db—ac 

~~      aba 

2abc 

bc  +  ab—ac' 

2abG 

■■—r- T-,  and  x= 

ab  +  ac—bc' 

2abc 

ac-\-bc—ab' 

adding  (3), 


Therefore. 


these  latter  vahies  bemg  written  out  at  once  hj  the  symmetry  of  the 
equations. 

Or  we  might  have  solved  the  problem  thus : 

Let  ic=the  number  of  units  of  work  performed  hj  A  in  one  day; 


Let  y=           " 

"             "      performed  by  B  in  one  day ; 

Let  z=z 

"             "     performed  by  C  in  one  day. 

hen, 

<^+y=^  (1); 

X  +  2=ry      (3); 

y+z=\    (8). 

These  give  by  eliminating,  as  before, 

bc-{-ab—ac 
7/  =  — -— . 

^  2abc      ' 

Therefore  J5's  time  of  performing  the  whole  work,  or  — =7 7 • 

^  °  y     bc  +  ab—ac 

as  before. 

198.  A  problem  may  often  be  solved,  as  readily  by  a  single 
equation  and  one  unknown  quantity,  as  by  simultaneous 
equations  with  two  or  more  unknown  quantities.  The  ad- 
vantage to  a  beginner  in  taking  several  letters  to  denote  the 
unknowns  is  that,  though  he  has  more  equations  and  longer 
work,  he  can  more  easily  follow  the  steps  by  which  the  equa- 
tions are  formed. 

Thus  Ex.  19,  Chap.  XXV.,  may  be  solved  by  four  simul- 
taneous equations,  involving  four  unknown  quantities. 


180  ELEMEJs^TARY    ALGEBRA. 


Examples — 41. 

1.  What  fraction  is  that,  to  the  numerator  of  which  if  7  be 
added,  its  yahie  will  be  | ;  but  if  7  be  taken  from  the  denom- 
inator its  value  will  be  |  ? 

2.  There  is  a  number  of  two  digits  wliich,  when  divided 
by  their  sum,  gives  the  quotient  4;  but  if  the  digits  change 
places,  and  the  number  thus  formed  be  increased  by  12,  and 
then  divided  by  their  sum,  the  quotient  is  8:  find  the 
number. 

3.  A  rectangular  bowling-green  having  been  measured,  it 
was  observed  that,  if  it  were  5  feet  broader  and  4  feet  longer, 
it  would  contain  116  feet  more;  but  if  it  were  4  feet  broader 
and  5  feet  longer  it  would  contain  113  feet  more:  find  its 
present  area. 

4.  A  person  rows  on  a  uniformly  flowing  stream  a  distance 
of  20  miles  and  back  again,  in  10  hours ;  and  he  finds  that 
he  can  row  2  miles  against  the  stream  in  the  same  time  that 
he  rows  3  miles  with  it.  Find  the  time  of  rowing  down  and 
the  time  of  rowing  up. 

5.  A  and  B  can  do  a  piece  of  work  together  in  12  days, 
which  B,  working  for  15  days,  and  C  for  30  days,  would  to- 
gether complete.  In  10  days,  working  all  three  together, 
they  would  finish  the  work :  in  what  time  could  they  sepa- 
rately do  it  ? 

6.  Some  smugglers  found  a  cave  which  would  just  exactly 
hold  the  cargo  of  their  boat,  viz.,  13  bales  of  silk  and  33 
casks  of  rum.  "While  unloading,  a  revenue  cutter  came  in 
sight,  and  they  were  obliged  to  sail  away,  having  landed  only 
9  casks  and  5  bales,  and  filled  one-third  of  the  cave.  How 
many  bales  separately,  or  how  many  casks,  would  it  hold  ? 


SIML'LTAXKOrS    IX^UATLO^^S.  181 

7.  Seven  years  ago,  A  was  three  times  as  old  as  B  was; 
and  seven  years  hence,  A  will  be  twice  as  old  as  B  will  be : 
find  their  present  ages. 

8.  A  certain  fishing-rod  consists  of  two  parts :  the  length 
of  the  npper  part  is  to  the  length  of  the  lower  as  5  to  7 ;  and 
9  times  the  npper  part,  together  Avith  13  times  the  lower  part, 
exceed  11  times  the  whole  rod  by  36  inches :  find  the  lengths 
of  the  two  parts. 

9.  If  the  nnmerator  of  a  certain  fraction  be  increased  by  1, 
and  the  denominator  be  diminished  by  1,  the  yalne  will  be  1 ; 
if  the  nnmerator  be  increased  by  the  denominator,  and  the 
denominator  diminished  by  the  nnmerator,  the  valne  will  be 
4:  find  the  fraction. 

10.  A  nnmber  of  posts  are  placed  at  equal  distances  in  a 
straight  line.  If  to  twice  the  nnmber  of  them  we  add  the 
number  of  feet  between  two  consecutive  posts,  the  sum  is  68. 
If  from  four  times  the  number  of  feet  between  two  consecu- 
tive posts  we  subtract  half  the  number  of  posts,  the  remainder 
is  68.     Find  the  distance  between  the  extreme  posts. 

11.  On  the  addition  of  9  to  a  certain  number  of  two  digits, 
its  digits  change  places;  and  the  sum  of  the  first  number 
and  the  number  thus  formed  is  33 :  find  the  digits. 

12.  A  and  B  ran  a  race  which  lasted  five  minutes.  B  had 
a  start  of  20  yards ;  but  A  ran  3  yards  while  B  was  running 
2,  and  won  by  30  yards:  find  the  length  of  the  course,  and 
the  speed  of  each. 

13.  A  person  has  two  casks,  with  a  certain  quantity  of 
wine  in  each.  He  draws  out  of  the  first  into  the  second  as 
much  as  there  was  in  the  second  to  begin  with;  then  he 
draws  out  of  the  second  into  the  first,  as  much  as  was  left  in 
the  first;  and  then  again  out  of  the  first  into  the  second,  as 
much  as  was  left  in  the  second.  There  are  then  exactly  8 
gallons  in  each  cask.     How  much  was  there  in  each  at  first? 


182  ELEMEIN^TAIIY    ALGEBRA. 

14.  The  year  of  our  Lord  in  which  the  ^  cltangc  of  style^ 
from  the  Julian  to  the  Gregorian  Calendar  was  made  in 
England,  may  be  thus  expressed :  The  first  digit  being  1  for 
thousands,  the  second  is  the  sum  of  the  third  and  fourth, 
the  third  is  the  tJiird  part  of  the  sum  of  all  four,  and  the 
fourth  is  the  fourth  part  of  the  sum  of  the  first  two.  De- 
termine the  year. 

15.  A  and  B  can  together  perform  a  certain  work  in  30 
days :  at  the  end  of  18  days,  howeyer,  B  is  called  ofi",  and  A 
finishes  it  alone  in  20  more  days.  Find  the  time  in  which 
each  could  perform  the  work  alone. 

16.  A  cistern  holding  1200  gallons  is  filled  by  three  pipes, 
A,  B,  C,  together,  in  24  minutes.  The  pipe  A  requires  30 
minutes  more  than  G to  fill  the  cistern;  and  10  gallons  less 
run  through  C  per  minute  than  through  A  and  B  together : 
find  the  time  in  which  each  pipe  alone  would  fill  the  cistern. 

17.  Find  two  numbers,  the  sum  of  which  is  equal  to  3 
times  their  difference,  and  their  product  equal  to  4  times 
their  difference.     (See  ISTote  at  the  end  of  "Examples — 40.") 

18.  The  sum  of  two  numbers  is  13,  and  the  difference  of 
their  squares  is  39.     What  are  these  numbers? 

Note. — Here  divide  the  second  equafion  by  tlie  first. 

19.  ^  and  B  are  two  towns,  situated  24  miles  apart,  on  the 
same  bank  of  a  river :  a  gentleman  goes  fi^om  A  to  B  m^ 
hours  by  rowing  the  first  half  of  the  distance,  and  walking 
the  second  half.  In  returning,  he  walks  the  first  half  at 
three-fourths  his  former  rate;  but  the  stream  being  with 
him  in  returning,  he  rows  at  double  his  rate  in  going ;  and 
he  accomplishes  the  whole  distance  in  6  hours :  find  his  rates 
of  walking  and  rowing. 

20.  Two  trains,  92  feet  long  and  84  feet  long,  respectively, 
are  moving  with  uniform  velocities  on  parallel  rails :  when 


SIMULTAXEOUS   EQUATIOKS.  183 

tliey  move  in  opposite  directions,  they  are  observed  to  pass 
each  other  in  one  second  and  a  half;  but  when  they  move  in 
the  same  direction,  the  faster  train  is  observed  to  pass  the 
other  in  six  seconds :  find  the  rate  at  which  each  train  moves. 

21.  A  raih^oad  runs  from  A  to  C.  A  freight  train  starts 
from  A  at  12  o'clock,  and  a  passenger  train  at  1  o'clock. 
After  going  two-thirds  of  the  distance  the  freight  train 
breaks  down,  and  can  only  travel  at  three-fourths  of  its 
former  rate.  At  40  minutes  past  2  o'clock  a  collision  occurs, 
10  miles  from  C,  The  rate  of  the  passenger  train  is  double 
the  diminished  rate  of  the  freight  train.  Find  the  distance 
from  A  to  C,  and  the  rates  of  the  trains. 

22.  A  certain  sum  of  money  was  divided  between  A,  B, 
and  C,  so  that  ^'s  share  exceeded  four-sevenths  of  the  shares 
of  B  and  (7  by  $30  ;  also  5's  share  exceeded  three-eighths  of 
the  shares  of  A  and  C  by  $30  ;  and  (7s  share  exceeded  two- 
ninths  of  the  shares  of  A  and  B  by  $30 :  find  the  share 
of  each  person. 

23.  A,  B,  and  C  can  together  perform  a  piece  of  work  in 
30  days ;  A  and  B  can  together  perform  it  in  32  days,  and 
B  and  C  can  together  perform  it  in  120  days  :  find  the  num- 
ber of  days  in  which  each  alone  could  perform  the  work. 

24  Express  the  two  numbers  whose  sum  is  a,  and  their 
diflPerence  h. 

25.  Find  two  numbers,  such  that  the  product  of  the  first 
increased  by  a,  and  the  second  increased  by  h,  exceeds  by  c 
the  produce  of  the  two  numbers ;  and  the  product  of  the  first 
increased  by  m,  and  the  second  increased  by  n,  exceeds  by  h 
the  product  of  the  two  numbers. 

26.  The  sum  of  two  numbers  is  a,  and  the  difference  of 
their  squares  is  h :  find  the  two  numbers. 

27.  Find  two  numbers,  whose  sum  is  m  times  their  differ- 
ence, and  their  product  is  n  times  their  difference. 


184  ELEMEJS^TARY    ALGEBJIA. 


XXVIII.    Indices. 


199.  The  indices  or  exponents  which  we  have  used  hith- 
erto are  positive  whole  numbers,  whicli  express  briefly  the 
repetition  of  the  same  factor  in  any  product.     (See  Art.  15.) 

Under  this  definition  we  have  proved  (Art.  59), 

Also  (Art.  74),  oT'-^cf—a'^'', 

200.  Hence  it  will  follow  that  (a'")^=:«"^"=(<^")^ 

For    {a;''y'  =  d!'\0r,ar\    &C.    n    factors   =^^+»»+"^+&c.  n  terms  ^^mn^ 

and  {o}'Y=^d!'.d!'.a\kQ>.    m  factors  ^z^'^+^+^+ac- ^ terms _^nm. 
.-.  since  a'^''~-=^cr\  we  have  {(f''Y=or'''—(a!'Y  \ 

that  is,  tlie  n*^  power  of  the  m*^  potver  of  2^  —  the  m*^  potoer 
of  the  Vl^ power  of  2.\  and  either  of  them  is  found  by  multi- 
plying the  two  indices. 

201.  Hence,  also,  it  will  follow  that     Va"^=(V<^)^ 
For  let    Va"'=:^^'^ ;        then  rr  =  {pf'Y^  {x^'Y  by  (199) ; 
hence         a  —  x^,  and  .*.  Va=2:,  and  {\/aY^^x'^\ 

but  also,  by  our  first  assumption,  V^"'^-^"'; 

hence,  we  have,      Va"'  =  (V^)"' ; 

that  is,  the  n*^  root  of  the  va^^  poiver  of  ^  —  the  m"'  poiver  of 
the  n*''  o^oot  of  a. 

202.  These  results  refer  so  far  to  positive  integral  indices. 
But  now  suppose  we  write  down  a  quantity  with  a  positive 
fraction  for  an  index,  and  agree  that  the  law  of  multiplica- 
tion, a^XaP'=a'^^'',  shall  hold  true  for  m  a7id  n  fractions  as 
well  as  m  and  n  j^ositive  ivhole  numbers.  What  would  such 
a  fractional  index  denote  ? 

For  example :  required  the  meaning  of  aK 

Indices.    Positive  fractional  indices. 


INDICES.  185 

By  supposition  we  are  to  liave  a'^Xa^—ci'—a.  Thus  a^ 
must  be  such  a  ii umber  that  if  it  be  multiplied  by  itself  the 
result  is  a ;  and  the  square  root  of  a  is  by  definition  such  a 
number;  therefore  a^  must  be  equivalent  to  the  square  root 
of  a,  that  is,  a7^=\/a. 

Again ;  required  the  meaning  of  a^. 

By  supposition  Ave  are  to  have, 

a^Xa^Xa^'--=a^-^^^"'^^=a'=a. 

Hence,  as  before,  a^  must  be  equivalent  to  the  cube  root 
of  fl^;  that  is,  a'^  =  ya. 

Again  ;  required  tlie  meaning  of  a*. 
By  supposition,        a^Xa^Xa'^X a*=a^ ; 
therefore,  a*  =  Va^. 

To  give  the  definition  in  general  symbols : 

1.  Required  the  meaning  of  a"  where  n  is  any  positive 
whole  numler. 

By  supposition, 

JL        -        JL  _L^  i-^J-_(. ...  to  n  terms 

a''Xa''Xa''X  ...  to  n  factors  —  a""    ""    ""  —a  =a; 

_i_ 
therefore  a"  must  be  equivalent  to  the  n^^  root  of  a, 

2_ 
that  is,  ar=Va. 

m 

2.  Required  the  meaning  of  a"  tuhere  m  a7id  n  are  any  pos- 
itive ivhole  numbers. 

By  supposition, 

!!L        i!i         !!L  ^',^.^..... ton  terms 

a'^Xce^Xa^'X  ...  to  ^z  factors  =  «j"    "    "  ==«"*; 

therefore  a''  must  be  equivalent  to  the  n^^  root  of  a^ ; 

VI 

that  is,  ^=Va"*. 


ISG  ELEMENTARY   ALGEBRA. 

m 

Hence,  a"  means  the  7^*^  root  of  the  m^^  power  of  a ;  that 
is,  in  a  fractional  index  the  numerator  denotes  a  power,  and 
the  denominator  a  root. 

203.  Again ;  if  we  wTite  down  a  quantity  with  a  negative 
index,  as  ar^  (where  p  is  either  an  integer  or  a  fraction),  and 
agree  that  this  symbol  shall  be  treated  by  the  same  law  of 
multiplication  as  a  positive  index ;  what  would  this  symbol 
denote  ? 

By  this  law  of  multiplication,  a^-^^  Xor^—d^''^^—dJ^\ 

but  we  have  also,  oJ^^^-^oF— — —  =  — '—-=d!^\ 

oF         oF  ^ 

so  that  to  multiply  by  a~^  is  the  same  as  to  divide  by  oF ; 

and  therefore, 

lXa-^=l-^«^  or  or''——-. 

Hence,  any  quantity  with  a  negative  index  denotes  the  re- 
ciprocal  of  the  same  quantity  with  the  same  positive  index. 

Thus  a~^=~,  a~^=r— ,  cri  =  -r=  — — ,  or=  >/a~^=  V— ; 
a  a  a^     Va  a 

^3     v/2  a 

Hence,  also,  any  power  in  the  numerator  of  a  quantity  may 
be  removed  into  the  denominator,  and  vice  versd,  by  merely 
changing  the  sign  of  its  index. 

Thus   a  ^trc  ^— 1=^-^= — ^=&c. 

c       cr^c      a 

204.  Lastly,  if  we  write  down  a  quantity  with  zero  for  an 
index,  as  aF,  and  agree  that  this  symbol  shall  be  treated  as 
if  the  index  were  ai;  actual  number— what  then  would  it 
denote  ? 

Since,  by  this  law,  a°Xtt"'=flf'+"'=«^^  it  follows  that  «°  is 
only  equivalent  to  1,  whatever  be  the  value  of  a. 

Negative  indices.    Zero  as  an  index. 


INDICES.  18? 

In  actual  practice,  such  a  quantity  as  a^  would  only  occur  in  certain 
cases,  where  we  wish  to  keep  in  mind  from  what  a  certain  number 
may  have  arisen :  thus  (»"  +  2d^-\-'da-\-&c.)-^d^=a-{-2  +  3a~^  +  &c.,  where 
the  2  has  lost  all  sign  of  its  having  been  originally  a  coefficient  of  some 
power  oia\  if,  hoAvever,  we  write  the  quotient  a  +  2«"  +  3«-^  +  &c.,  we 
preserve  an  indication  of  this,  and  have,  as  it  were,  a  connecting  link 
between  the  positive  and  negative  powers  of  a. 

The  quantity  a*  is  still  called  a  to  tlw  power  of — ,  and  similarly  in  the 

case  of  a-^^^  a^ ;  but  the  word  power  has  here  lost  its  original  meaning, 
and  denotes  merely  a  quantity  with  an  index^  whatever  that  index  may 
be,  subject,  in  all  cases,  to  the  Law,  a"^.  a^  =  a^+^ . 


Examples — 42. 
Express,  with  fractional  indices, 

1.  Vx'  +  Vx'-h{^xY+{Vxy;  V{a'b')^V{a'b')-{-V{ab') 

+  V{a'b'). 

2.  aVb'  +  {s/ay  +  V{a'b)-i-V{a'b');  V{aW)+a{Vby 

+V{a'b'')+V{a'b'y 

Express,  with  negative  indices,  so  as  to  remove  all  powers, 
1st,  into  the  numerators,  and  2d,  into  the  denominators. 

1      2      3      4«     5Z>    «^     U'     6a    U     2F 

^*    '^'^b'^'7'^'b^/a''  J^^T^T'-^^'^-^' 


3^V    a'b'    a  ^ Zabc'  %y c^ ?> sf a'^ ^V {a'bc'y  d^^/W 
Express,  with  signs  of  Evolution, 

JL  3.7  X         ^    1.     3 

5.     «^  +  2r/  3  4-  3^  4  ^  4^  6  4.  ^4 ,  _|.  . _| — 

b'         2c  ^  db^ 


J)l  c^       b^ c^ 
4^^  6a^ 


Express,  with  positive  indices,  and  with  the  sign  of  Evo- 
lution, 


188  ELEMENTARY     ALGEBRA. 

+a-H^  +  b-^' 

cr%^      2a_    3b-'c-'    1 ^    ^    ^    ^ 

^'       C-'    '^b-'c-'^     a-'    '^a-'b-'c-''  ^'k'^a-i'^a-^^  ^~"' 
205.  From  our  definitions,  {a^  y  =  a^  said  (a^)'*  =  a's   and 
in  general,  (a «)""  =  {a'"'') n ;  then,  also,  [a'^)^  =  a^^ -^  also,  a^, 

bn  ,c^  , , ,  z=i[abc  .  . .  )"^  since  each,  raised  to  the  n^'^  power, 
gives  abc. 

It  follows,  then,  that  ^vJiatever  be  the  indices, 
cd"^  1 

It  will  be  observed,  also,  that  since  a  fraction  may  take 
different  forms  without  changing  its  value,  the  form  of  a 
fractional  index  may  be  changed  without  altering  the  valuo 
of  the  quantity. 

Thus  (3^  3  —  ^(j  •  for  either  raised  to  the  sixth  power  gives  a* ; 

and  in  general,  a^=a^,  for  either  raised  to  the  np  power 
gives  a"'^.  Hence  we  can  reduce  the  indices  of  two  quantities 
to  a  common  denominator  without  changing  the  value  of  the 
expressions. 

20^.  Hence,  (1)  to  multiply  together  any  powers  of  the  same 
quantity  we  must  add  the  ijidices  ;  (2),  to  divide  any  one  power 
of  a  quantity  by  another  power  of  the  same  quantity,  subtract 
the  index  of  the  divisor  from  that  of  the  divideiid ;  and,  (3), 
to  obtain  sluj  poiuer  of  a  poiver  of  a  quantity,  we  must  muIM- 
ply  together  the  two  indices. 

The  signs  of  the  indices  must  of  course  be  carefully  ob- 
served. 

Ex.1.  Multiply    a^'b^ c^hj  a^'b^c^ . 

Reduction  of  the  indices  of  two  quantities  to  a  common  denominator.  Rule  for 
(1)  multiplying  together  powers  of  the  same  quantity ;  (2)  for  dividing  one  power  of  a 
quantity  by  another  power  of  the  same  quantity ;  (3)  for  obtaining  any  power  of  a 
power  of  a  quantity. 


IXDICES.  189 


3  "^2  ""6'  4'^8~12'  3'^3~    * 


therefore,  c^  h^  c  «  x r^^  Z>  -^  t^  ^  =  ^^  b ^^  ^• 

Ex.2.    Divide      x^  y^^  hj  x^^  y'^ , 

4       2~4'  3       6~2' 

tlierefore,  ir*  y-^  -^x'  y^  =zx^  y~ . 

Also, 

a^Xa-^-:a^~'=^;  «^--«-*=:a^+^^ri^;  a-^-4-a-^=a-*+^=a-^'-<^ ; 
Ex.  3.    Multiply     x-\-x^  -\-  x~^  by  cc^  +  i?;"^  —  x~^. 


X   -\-x'^ 

+  x-i 

x^  ■\-x~^—x-'- 

• 

x^+x^ 

+  1 

x^ 

+  1+x 

4 

-  1-x 

-I. 

-x-i 

x^^lx- 

i+l 

-x^ 

Here  in  the  first  line,  x^  Xx=x^^^^  —  x^^\  x^  Xx^  =  x^ \ 
x'^  X  X  '^=x'^  =  l]  and  so  on. 

Ex.  4.   Divide 

x^  -  a^  x'-Ux^  ^eJx~  2a V'  by  x^  -4.ax^  +2  J  • 

x^  -  4:ax^  +  2a^ )  x^  -  a^  x'  -  4.ax^  +  6a^  x-2a''x^  (x-a^  x^ 
x^  —  4:ax^  +  2a^  x 

-a^x'  -h  4:J x-~2a''x^ 

-a^x'  -h  4:a^  x-2a*x^ 


190  elemektaky  algebka. 

Examples — 48 
Find  the  value  of 

1.  9-^.       2.  4~i      3.  (100)-^.      4.  (1000)^.      5.  81"^. 
Simplify 
6.     (a=)-l  7.     (a-^)-^  8.     ^/a-^  9.     V^"". 

10.  aixaixa~^' 
Multiply 

11.  X"^  +  ^4   ]3y  ^f  _  ^!  ^ 

13.  x-\-x^'  +  2  by  iz;  +  :2;*  -  2. 

14.  .^*+:^'  + 1  by  ^'c-'-ii;-'  + 1. 

15.  oT^^ar'-^l  by  a~i^l. 

16.  «3_2+^-^  by  a^-^-t 

17^    a  +  ^^Z**  — ii;"^^/^  by  a  +  a^^i+o;*^^. 

Divide 

18.    :^;t-3^t  byi^;*-^*.  19.     a-5bya^-J^. 

20.  64^-^  +  27y-^  by  4:?;-^  +  Zy-^\ 

21.  o;^  —  x]p  +  i^^?/  -  ^^  by  o^  —  ^-i. 
22..    a^  ^-a^l^  ^ifih^  a^  ^a^lh  j^l\^ 

23.  fl^^  +  ^>5_c§ +2^3^*  by  a^  +  ^^  +  c^. 

24.  ^^  -  '^a^x^  +  a'  by  x^  -  %a^  x^  +  «. 

Find  the  square  roots  of  the  following  expressions. 

25.  a:* -4  +  42;-*.  26.     (a:  +  aj-^)2-4  (i?;~ar-^). 
27.    a^l-''  +  2a^-^  4-  3 + ^ar^l  +  a-^Z>l 


SURDS.  191 

XXIX.    Surds. 

207.  It  was  stated  (Art.  147)  that  when  any  root  of  a  quan- 
tity cannot  be  exactly  obtained  it  is  expressed  by  the  use  of 
the  sign  of  Evolution,  as  \/5,  V{3ab),  \/(a'  +  c'');  and  such 
quantities  are  called  Iri^ational  quantities,  or  Surds. 

It  is  also  stated  in  (147)  that  there  cannot  be  any  even 
root  of  a  negative  quantity,  but  that  such  roots  may  be  ex- 
pressed in  the  form  of  surds,  as  \/  — 3,  V— Z>^  V—{a^-\-lf)y 
and  are  then  called  impossible,  or  imaginary,  quantities. 

208.  Since  every  fractional  index  indicates  by  its  denomi- 
nator a  root  to  be  extracted,  all  quantities  having  such  in- 
dices are  expressed  as  Surds. 

When  a  negative  quantity  has  the  denominator  of  its  in- 
dex (reduced  to  its  lowest  terms)  even,  the  expression  will  be 
imaginary. 

Thus,  (—3)^,  (  —  9)4,  are  imaginary  quantities;  but 
(— 4)ti  is  not  so,  since  it  is  the  same  as  (  —  4)^,  where  the 
root  to  be  taken  is  odd, 

209.  The  operations  of  the  preceding  chapter  are  opera- 
tions on  surds,  but  we  may  apply  the  rules  which  are  de- 
monstrated in  that  chapter  by  the  use  of  fractional  indices, 
also  to  surds  expressed  by  the  sign  of  Evolution,  or  Radical 

Sign, 

2i0.  In  the  case  of  a  mmierical  surd  expressed  with  a 
fractional  index,  should  the  numerator  be  any  other  than 
unity,  we  may  take  at  once  the  required  power,  and  so  have 
unity  only  for  the  numerator,  and  simply  a  root  to  be  ex- 
tracted. 

Thus,  2^  =  {2')^  =  4*  =V4;  3-^=:(3-^)*  =  (^)^  =V^S, 

Surds.    Surds  expressed  by  the  radical  sign. 


192  ELEMENTARY     ALGEBRA. 

211.  Quantities  are  often  expressed  in  the  form  of  surds, 
which  are  not  really  so — i.e.,  when  we  ca7i,  if  we  please,  ex- 
tract the  roots  indicated. 

Thus,  Va,  Vh  {cc'  +  ah  +  h'')^  are  actually  surds,  whose 
roots  we  cannot  obtain;  but  Va^,  \/27,  (4a^-|-4«Z>  +  ^^)J,  are 
only  apparently  so,  and  are  respectively  equivalent  to  a,  3, 
2a-\-l). 

Conversely,  any  rational  quantity  may  be  expressed  in  the 
form  of  a  surd,  by  raising  it  to  the  power  indicated  by  the 
root-index  of  the  surd. 

"For  example,    3-=  v/3'r:.  v/9  ;   4=V4'  =  V64; 
a^\/a'\  a  +  h=:l/{a-\-l)y. 

212.  In  like  manner  a  mixed  surd,  i.  e.  a  product  partly 
rational  and  partly  surd,  or  a  surd  with  a  rational  coefficient, 
may  be  expressed  as  an  entire,  i.  e.,  complete  surd,  by  raising 
the  rational  factor  to  the  power  indicated  by  the  root-index 
of  the  surd,  and  placing  beneath  the  sign  of  Evolution  the 
product  of  this  power  and  the  surd  factor.  An  entire  or 
complete  surd  is  one  in  which  the  whole  expression  is  under 
the  sign  of  Evolution. 

Thus, 

2v/3=v/4Xv/3=v^l2;  3. 23=3^4^ V27x  1/4 rr:Vl08  ; 

iCa  /Ca 

Conversely,  a  surd  may  often  be  reduced  to  a  mixed  form, 
by  separating  the  quantity  beneath  the  sign  of  Evolution 
into  factors,  of  one  of  which  the  root  required  may  be  ob- 
tained, and  set  outside  the  sign. 

Thus,    v/20=:n/(4x5)  =  2v/5;  V24=V(8x3)==2V3; 
y/{^a'h)^^a>/{ZaV) ;     \/{^a'h'c')=iahV(2ac'). 

Rational  quantities  in  the  form  of  a  surd.    Mixed  surds  and  entire  surds. 


SUKDS.  193 

213.  A  surd  is  reduced  to  its  simplest  form,  when  the 
quantity  beneath  the  root,  or  surd-factor,  is  made  as  small  as 
possible,  but  so  as  still  to  remain  integral. 

Hence,  if  the  surd-factor  be  a  fraction,  its  numerator  and 
denominator  should  both  be  multiplied  by  such  a  number  as 
will  allow  us  to  take  the  latter  from  under  the  root. 

Thus, 

/2         /2.3      1    ,^     5    s /24       ^  ;/3        ,  f/3.5^     ,,^. 


/3^    /3X2^    /_^^v^6 
r    ft     r  ftv2     y  Ifi 


8X2     ^  16       4   ' 

y  2  _  ;/^X9_  yi8_  V18 
^  3-'^3x9~'^27~    3    * 

214.  Surds  which  have  not  the  same  index  may  be  trans- 
formed into  equivalent  surds  which  have.     (See  Art.  204.) 

For  example,  take  \/5  and  Vll ; 

v/5  =  5^,  Vll  =  (ll)3  ; 

5^  =5^  =  V5^= V125,  (11)3  =111  =V(11)'=V121. 

215.  Similar  surds  are  those  which  have,  or  may  be  made 
to  have,  the  same  surd-factors. 

•  Thus  3  Va  and  Va,  "^aVc  and  ?>Wc,  are  pairs  of  similar 
surds ;  bVa  and  '^VaJ^  are  similar  surds,  because  ^^/a"  may  be 
written  2 V^^;  and  \/8,  \/50,  \/18  are  also  similar,  because 
they  may  be  written  2v/2,  5\/2,  3\/2. 

Examples — 44. 

1.    Express  4^,  9^,  3~s  2"^,  (f)"^,  (^)-^,  with  indices, 
whose  numerator  is  in  each  case  unity. 

Eeduction  of  surds  lo  their  simplest  form,  —to  equivalent  surds  having  a  given 
index.    Similar  surds. 

9 


194  ELEME]S^TAIIY    ALGEBRA. 

2.  Express  5,  2^,  |«,  ^a'^,  2(^^  +  ^)^  ^^  surds,  Avith  indices 
I  and  I". 

3.  Express  d~%  (3^)"^  a~\  ab~'c~\  with  indices  ^  and 

Eeduce  to  complete  or  entire  surds, 

4.  5v/5,  2Vh  |.3S  fN/li  i(|)-S  25(li)-^. 

5.  dV2,  8X2-S  4X2^  3x3-^  Kf)"',  id)"*. 

6.  2v^^,  7a%/(22:),  (^  +  Z^)  (^^-Z,^)-i. 

^     a  ^  3x    3b  ^   2a    3  ^   4a'     ^         ^  '^  a-\-x 

Eeduce  to  tlieir  simplest  form, 

8.  v'45,x/125,  3v/432,  V135,  31/432,  v/f,  2V|,  3Vi  4V3-|. 

9.  si,  32%  72*,  (H)-^,  (20i)-t,  (30|)-^  fv/-^^,  5V4^V, 

10.  Show  that  x/12,  Sv'TS,  |V147,  fV^.  VyV  and  (144)-i 
are  similar  surds. 

216.  To  compare  surds  with  one  another  in  magnitude, 
express  tliem.  as  entire  surds,  and  then  reduce  them,  if  neces- 
sary, to  a  common  siird-index,  and  simplify  as  in  Art.  213. 
Their  relative  values  will  then  be  apparent. 

Thus,  3\/2  and  2\/3,  expressed  as  entire  surds,  are  \/18  and 
\/12,  and  it  is  at  once  plain  which  is  the  greater. 
To  compare  \/5  and  VH  : 

v/5-:5^=5^r=V125; 
V11  =  11^=::11*=n/121. 
We  see  now  that  \/5  is  greater  than  Vll. 

217.  To  add  or  subtract  similar  surds,  reduce  them,  luhen 
necessary,  to  the  same  surdf actor,  then  add  or  subtract  their 
rational  factors  or  coefficients,  and  affix  to  the  result  the  com 
mon  surd-factor. 

>  compare  surds  with  one  another  in  magnitude.    To  add  or  suhtract  similar  surds. 


SUEDS.  •  195 

Thus,      v/8+v/50-v/18  =  2\/2  +  5v/2-3N/2:rr4v/2; 
^aV{a'h')-\-bV{^a'h)-V{nba'h')  ^A.a'Wl  +  WbVh-^ 

hcCWl^a^Wb. 
2'  Id      1'   7256     2'  /12     1'  /64xl2 


2'  /3      1'  /256     2'  /12     1'  /6 


27 


__2_V12     1  4V12_2V12 
■~3     2    "^4      3     ~"    3     • 

Dissimilar  surds  can  be  added  or  subtracted  only  by  con- 
liccting  them  with  their  proper  signs. 

218.  To  midtiply  simple  surds  which  have  the  same  surd- 
mdex,  multiply  separately  the  rational  factors  and  the  surd 
factors,  retaining  the  same  surd-index  for  the  product  of  the 
latter. 

Thus,        3v/2X\/3  =  3v/6;  4\/oX7v/6=:28n/30; 

2V4X3V2=:6V"8  =  6X2  =  12; 

2v/3x3n/10x4v/6  =  24v/180  =  144\/5. 

219.  To  multiply  simple  surds  which  have  not  the  same 
surd-index,  reduce  them  to  the  same  surd-index  and  proceed 
as  hefore. 

Thus, 

4v/5x2V11  =  4V125x2V121:=8V(125x121)  =  8V15125; 

2^3  X  3  V2 =2V27  X  3  V4= 6V108. 

Compound  surd-expressions  are  multiplied  according  to 
the  method  of  compound  rational  expressions. 

Ex.  1.     (2±N/3)=^=:4dz4v/3  +  3  =  7±4y3. 

Ex.  2.     (2+  v/3)  (2-  v/3)  =  4-3  =  l. 

Ex.3.     (2+v/3)(3-\/2).=  6  +  3v/2-2v/2- v/6. 

To  multiply  simple  surds— two  cases. 


196  ELEMENTARY    ALGEBIIA. 

220.  To  divide  one  simple  surd  by  another^  reduce  hotli 
surds  to  the  same  surd-index,  ivlien  necessary ;  then  divide  the 
coefficients  and  surd-factors  separately,  retaining  the  common 
surd-index  aver  the  quotient  of  the  latter. 

The  result  may  be  simplified  by  Art.  212. 


—  o  f 


125  X 121 X 121     4V1830125 


"3"    121X121X121         3X11 
Ex.3,     (8^/2-12^/3  +  3^/6-4)-^2v/6 

=:4v/|~6v/|  +  |--^=tv/3-3x/2+|-ix/6. 

Ex.  4.     (2v/3-6V2)-f:\/6=:2v/|-6V-^t6  =  %/2-V864. 

221.  But,  if  the  divisor  be  not  monomial,  the  division  is 
not  so  easily  performed.  The  form,  however,  in  which  com- 
pound surds  usually  occur,  is  that  of  a  l)inomial  quadratic 
Burd,  i,  e.  a  binomial,  one  or  both  of  whose  terms  are  surds, 
in  which  the  square  root  is  to  be  taken,  such  as  3  +  2\/5, 
2v/3— 3v/5,  or,  generally,  s/a^s/h,  where  one  or  both  terms 
may  be  irrational;  and  it  will  be  easy,  in  such  a  case,  to 
convert  the  operation  of  division  into  one  of  multiplication, 
hy  putting  the  dividend  and  divisor  in  the  form  of  a  fraction, 
and  multiplying  loth  numerator  and  denominator  by  that 
quantity  which  is  obtained  by  changing  the  sign  between  the 
ttvo  terms  of  the  denominator.  By  this  means  the  denomina- 
tor will  be  made  rational:  thus,  if  it  be  originally  of  the  form 
\/a±  \/Z>,  it  will  become  a  rational  quantity,  a—b,  when  both 
numerator  and  denominator  are  multiplied  by  Va^i  y/b. 

To  divide  one  Bimple  surd  by  another.    To  divide  binomial  quadratic  surds. 


SUKDS.  197 

2+v/3_(2+v/3)  (3-n/3)_6+3v/3-2v/3-3 
•     3+v/3~(3+v/3)(3-v/3)~  9-3 

3+V/3 


Ex.2. 


6 
2\/2+\/3     2v/2+v/3 


2v/2~v/3~      8-3  5 

This  process  is  called  rationalizing  the  denominators  of  the 
fractions,  and  the  fractions  thus  modified  are  considered  to 
be  reduced  to  their  simplest  form. 

Examples — i.5, 

1.  Compare  6v/3  and  4v/7;  3V3  and  2V10;  2V15,  4V2, 
and  3V5. 


Simplify 

2.  3v/2  +  4:n/8-v/32. 

3. 

2V4:+5V32-V10« 

4.  2v/3  +  3v/(li)-v/(5i). 

5. 

1          1 

V2  ~  V16' 

Multiply 

-1          -j 

6.   v/5+%/(li)--;^by  v/3. 

7. 

'^^     Vl6+V3^y'^^- 

8.  l+%/3-v/2by  v/6-v/2. 

9. 

v/3+v/2byj3H--i. 

10.  Divide  * 
2x/3  +  3v/2  + V30  by  3x/6,  and  2^/3  +  3V2+V30  by  3^/2 

11.  Eationalize  the  denominators  of 

__1 4  3 8--5v^        3  +  \/5 

2^/2-^/3'       Vb-1'       \/5  +  v/2'       3-2v/2'       3-n/5' 

12.  Eationalize  the  denominator  of  —7 "-4: 77 7. 

V{a-\-x)  —  y/{a—x) 

222.  The  squar®  root  of  a  binomial,  one  of  whose  terms  is 
a  quadratic  surd  and  the  other  rational,  may  sometimes  be 
expressed  by  a  binomial,  one  or  both  of  whose  terms  are 

Eationalizing  the  denominators  of  fractions.    To  express  the  square  root  of  a 
Dinomial,  one  of  whose  terms  is  a  quadratic  surd  and  the  other  rational. 


198  ELEMEKTAKY    ALGEBKA. 

quadratic  surds.     (A  quadratic  surd  is  one  whose  indicated 
root  is  the  square  root.) 

Since  {y/x:^Vyy=^x-±:^Vxy  +  y,  therefore  Vx^2Vxy-\-y 
=z^xdci^y'^  hence  if  any  proposed  binomial  surd  can  be  put 
under  the  form  x±^Vxy  +  y,  its  root  may  be  found  by  in- 
spection to  be  \/x±^/y.  To  show  how  to  proceed  in  any 
proposed  case,  let  us  take  the  binomial  3  +  2\/2.  To  place 
this  under  the  form  x  +  2y/xy-\-y,  we  observe  that  2\/2 
=2v'2X\/l,  and  the  sum  of  the  two  numbers  under  the 
radical  signs  2  +  1=3,  the  rational  term  of  the  binomial; 
/.  34-2\/2=2  +  2\/2X\/l  +  l.  Hence  the  square  root  of 
3+2v/2='v/2  +  2V2X\/l  +  l=:\/2  +  l. 

Ex.  2.     Kequired  the  square  root  of  7—  2\/10. 

Here2yiO=2v/5X\/2. 

Also,  5  +  2=7,  the  rational  term.      Hence, 
7-2x/10=5~2v/5X\/2  +  2;     .-.  root  required  is  Vo-V2. 

Ex.  3.  Eequired  the  square  root  of  11  — 6\/2. 

Here  6x/2=2v/18=:2v'9X\/2,  or  2v/6X\/3;  of  which  the 
former  answers  the  condition  that  their  sum  9+2=11,  the 
rational  term;    .-.  ll-6y2=:9--2x/9x%/2  +  2. 

A  the  required  root  is  \/9  — \/2;  that  is,  3  — %/2. 

These  illustrations  show  the  method  to  be,  to  put  the  term 
loMcli  contains  the  surd  into  factors,  of  the  form  ^VxXVy, 
in  such  a  manner  that  the  sum,  x-{-y,  of  the  numbers  under 
the  ttvo  radicals  may  be  equal  to  the  rational  term.  Then  the 
y/x'±.\/y  loill  be  the  required  square  root, 

Ex.  4.     Eequired  the  square  root  of  7  +  \/13. 
Here  N/13  =  2v'y  =  ^^^¥X^/i. 
Also,  1^  +-|-=7,  the  rational  term. 

%/13+l 


/.  root  required  is  x/V  +  v^i,  or 


n/2 


SURDS.  199 

Examples — 46. 
Find  the  square  roots^  of 

1.  4+2x/3.      2.  ll^-6^/2.      3.  8-2v/15.      4.  38-12v/10. 
5.  41-24v/2.        6.  2-1— v/5.        7.  4|-fv'3. 

2222^.  It  is  often  required  to  c/mr  an  equation  of  surds.  An 
equation  may  be  cleared  of  a  surd  by  transposing  the  terms 
so  that  the  surd  shall  form  one  side  and  the  rational  quanti- 
ties the  other  side,  and  then  raising  loth  sides  to  that  poiver 
which  will  rationalize  the  surd. 

In  the  case  of  quadratic  surds  we  square  both  sides.  We 
shall  confine  ourselves  to  clearing  equations  of  quadratic 
surds. 

Thus  if  ^ya^-x—h  —  c,  by  transposition, '^«  +  :i'=:^  +  c;  and, 
squaring  both  sides,  a  +  x—{l)-\-cy.  We  thus  have  an  equa- 
tion without  surds. 

If  the  equation  contain  tivo  surds  connected  by  the  signs 
+  or  — ,  then  the  same  operations  must  be  repeated  for  the 
second  surd. 

Thus,  if  "^a  +  x  ^s/x—l, 

by  transposition,  '^ a-\-x=l)—\/x\ 

squaring,  a-^-x^y^—^bs/x-^rX^ 

reducing  and  transposing,     %b\fx—lf—a\ 
squaring,  W'x^iV^  —  aY, 

an  equation  in  which  the  surds  do  not  appear. 

Ex.  1.  ^b-\-x-\-  "^b—x—2\/x\  required^. 
Transposing,  ^b-\-x=-2Vx —  ^b—x  ; 

squaring,  b-\-x—A:X—4:^bx—x''-\-b—x] 

To  clear  an  equation  of  surds. 


200  ELEMF.XTAllY     ALGEBKA. 


reduc'g  and  transp'g,   4:V5x—x^—2x; 
%y/hx—x^=  x; 
squaring,  20x—^x^=x* ; 

20x=bx'', 
.'.  20= 5a;,  and  a;=4. 

Examples— 47. 

Find  X  in  the  following  equations : 

1.  ^/(4a;)+v/(42;-7)  =  7.         2.   V{x-\-14.)+V{x-U)=lL 

3.  x/(a;  +  ll)4-\/(a;-9)  =  10. 

4.  v/(9a;+4)  +  v/(9a;~l)=3. 

5.  \/(a;+4aZ>)=2a— \/:r. 

6.  v/(a;-a)+v/(2;-Z>)=x/(a~^>). 

7.  a+a;~N/(2a:?;  +  ir')=&.  8.  a-{-x^-y/{a^  +  'bx  +  x^)=b. 

XXX.    Quadratic  Equations. 

223.  Quadratic  equations  are  those  in  which  the  square 
of  the  unknown  quantity  is  found,  but  no  higher  power  of 
it.    Of  these  there  are  two  species : 

1.  Pure  Quadratics,  in  which  the  square  only  of  the  un- 
known is  found  without  the  first  power,  as 

a;'-9=0;         ic'-a'=Z^^  &c. 

2.  Affected  Quadratics,  where  the  first  power  enters  as 
well  as  the  square,  as 

a;'— 3a;  +  2=:0;         ax^-\-hx^c\  &c. 

224.  Quadratic  equations  are  (Art.  1G6)  also  called  Equa- 
tions of  tlie  second  degree.     The  two  species  also  are  distin- 

Quadratic  equations ;  two  species  of,  and  two  ways  of  designating  them. 


QUADEATIC    EQUATI0:N'S.  201 

guished  as,  1st,  Incomplete  equations  of  the  second  degree ; 
and  2d,  Complete  equations  of  the  second  degree.  We  shall, 
however,  generally  use  the  notation  of  Art.  223.* 

225,  To  solve  a  Pure  Quadratic  Equation : 

Find  tlie  value  of  x^  ly  the  ride  for  solving  simple  equa- 
tions ;  then  take  the  square  root  of  both  sides  of  this  result, 
we  thus  find  the  value  of  x,  to  which  we  must  prefix  the 
douUe  sign  =b  (144). 

Such  equations  will  therefore  have  two  equal  roots  with 
contrary  signs. 

Ex.  1.  x^-^=^. 

Here  a:^=9,  and  x=^^. 

If  we  had  put  db:?;=:±3,  we  should  still  have  had  only 
these  two  different  values  of  a;,  viz.,  a;  = +3,  a;=— 3;  since 
—x—^'d  gives  x=—d,  and  —x=—^  gives  x=  +3.       * 

Ex.  2.'         ^(3:?;=  +  5)-i(a;''  +  21)  =  39-5a;'. 
Reducing,  121a;'^=:1089; 

.*.  x^=^,  and  ^=±3. 

a;^  +  2      9 


Ex.  3. 


a;^-2~  7' 


To  examples  like  this  the  principle  of  fractions,  (Art.  134, 
vi),  may  be  applied  with  advantage  when  the  unknown 
quantity  does  not  enter  in  both  sides  of  it. 


*  The  term  affected  was  introduced  by  Vieta,  about  the  year  1600. 
It  is  used  to  distinguish  equations  which  involve  or  are  affected  with 
different  powers  of  the  unknown  quantity  from  those  which  contain 
one  power  only.    (Lund.). 

To  Boh'e  a  pure  quadratic. 


202  ELEMEJS^TARY    ALGEBRA. 

{x'-i-2)-lx'-2)~9-7' 


that  is, 


2^_16 

T'~~2'' 


Ex.  4.  - — =4. 


ir"— 16,  and  x—±4c. 


(134,  Yi) 

V4:  +  X'        5 

X          3' 

squaring, 

4:  +  x'     25 
x'    -9' 

again,  (134,  iy  and  i). 

x'      9           ,9             ,  , 
-_-;   r,x--;    a:-±f. 

Examples — 48. 

1.  ix'=U-3x\ 

2.  x'  +  5=^x'--16. 

3.  (a;+2)^=4^  +  5. 

'■  it+i-.-»- 

3        17 

4.x'     6x'~3' 

a;       7 

dx'      15a;'' 4- 8 
^'4             6      ~ 

2x' 

,        ^   x^      a;' -10     ^     50+a? 
^-       ^-   5          15     =^        26 

.    3:2;*-27     90+4^;^     ^  Ar^^'  +  S      2.^^-5     7ii;'^-25 

^'  _2     ,     O       + 2     .     A       ~7.  lU. 


11. 

12. 


a;''^+3   ^  aj'-h9  '10  15     ~      20 

10:r^  +  17  _  12^N-2  _  5a;'' -4 
18       ■~lla;''-8~       9~"' 

140;"  + 16       2a;'' +  8  ___  2^ 
21       ~8a;''-ll~'3"' 


QUADKATIC    EQUATjg^is^S.  203 

2 2 

^16  +  af-'^25^^_l_ 

226.  An  affected  quadratic,  or  comiDlete  equation  of  tlie 
second  degree^,  may  alivays  he  reduced  to  the  form  x^  +  px  +  q 
=  0,  ivliere  the  coefficient  of  x^  is  +1,  and  p  and  q  represent 
known  numbers,  luliole  or  fractional,  positive  or  negative. 

For,  let  all  the  terms  be  brought  to  one  side,  and,  if  neces- 
sary, change  the  signs  of  all  the  terms,  so  that  the  coeflScient 
of  ^  may  be  a  positive  number ;  then  divide  every  term  by 
this  coefficient,  and  the  equation  takes  the  assigned  form, 
X'  -^px-{-q—^. 

jN'ow  in  this  equation  we  have  x^ ^px——q',  and  adding 
(kvf  ^^  ^^^^  side,  we  get  '^^ ^-px'\-\p' =\p^ —q\  by  this  step 
the  first  side  becomes  a  complete  square  (Art.  153);  and 
taking  the  square  root  of  each  side,  prefixing,  as  before,  the 
double  sign  to  that  of  the  second  side,  we  have 

x-V\p^±^W^\ 

which  expression  gives  us,  according  as  we  take  the  upper  or 
lower  sign,  two  roots  of  the  quadratic. 

227.  From  the  preceding  we  derive  the  following  rule  for 
the  solution  of  equations  containing  an  afiected  quadratic : 

By  reduction  and  transposition  arrange  the  equation  so 
that  the  terms  involving  x^  and  x  ai^e  alone  on  one  side,  and 
the  coefficient  of  x"  is  + 1 ;  then  add  to  each  side  the  square  of 
half  the  coefficient  of  x,  and  talce  the  square  root  of  each  side, 
prefixing  the  double  sign  to  the  second. 

To  what  form  may  an  affected  quadratic  always  be  reduced  ?  Rule  for  the  solution 
of  equations  containing  an  affected  quadratic  ? 


204  ELEMENTARY    ALGEBRA. 

We  thus  obtain  a  simple  equation  from  which  x  is  readily 
found. 

Ex.  1.  x'-6x==7. 

Here  x'-ex-}- 9=^7 +  9  =  16; 

whence  ^—3  =  ±4, 

and  a;=3+4=7,  ori^=3-4=-l; 
so  that  7  and  —1  are  the  two  roots  of  the  equation. 

Ex.  2.  x''-i-Ux=95. 

Here  a;' +  14a; +  49=:  95 +  49  ==144; 

whence  a; +  7=  ±12, 

and  ir=  — 7±12=5,  or  —19. 


Examples — 49. 
Solve 

1.   x''-2x=8.  2.   a:'  +  10:?;=-9.  3.   x^'-Ux^VZO. 

4.   a;''-12a;=-35.      5.   a;'  +  322;=320.     6.   i^;'  + 100a; =1100. 

228.  If  the  coefiicient  of  x  be  odd,  its  half  will  be  a  frac- 
tion. Its  square  may  be  indicated  on  the  first  side  bv  using 
brackets. 

Ex.1.   •  x'-6x=z-6. 

Here  x''-5x+{iy= -6  +  ^=1; 

whence  a;— 1=±|, 

and  a;=|  +  |=f  ==3,  or  x=^—^=^=2, 

Ex.  2.  x'-'X=l. 

Here  x'-x+{iy=i+i=l; 

whence  cc— ^=±1, 

and  a;=i+l=l|;   ora;=J— 1  =  — ^. 


quadratic  equations  s.  205 

Examples — 50. 
iSolye 

1.  ^'  +  7a;=8.  2.  ^j'^-IS^j^GS.      3.  x'  +  2dx=-100. 

4.  i?;'  +  13ic=-12.      5.  a;' +  190:= 20.       6.  cc'  + 1112:= 3400. 

229.  If  the  coefficient  of  a;  be  a  fraction,  its  half  will,  of 
course,  be  found  by  halving  the  numerator,  if  possible;  if 
not,  by  doubling  the  denominator. 

Ex.1.  Solve    x'  +  y^x=19. 

Here  x'  +  \^x+{iy=19-i-^^=^^; 

whence  a;+|=±^, 

and  a;=  — 1  +  ^=3,  or  x=—^—^='-6^, 

Ex.2.     Solve    x'+^x=U, 

Here  x'+h'-x+my=U-hm='^'^y 

whence  x+\^=±^^y 

and  x=-{i+U=n,  or  ^=_^— 1-|=-10. 

♦ 

Examples— 51. 
Solve 

1.  x'-ix=d4..  2.  x'-ix=27,  3.  x'+^xz=86. 

4.  x'-^x=lU.        5.  x'-{-j\x=U6,        6.  ^'^ -f|:z;=147. 

230.  In  the  following  examples  the  equations  will  first  re- 
quire reduction ;  and  since  the  rule  requires  that  the  coeffi- 
cient of  x^  shall  be  + 1,  if  it  have  any  other  coefficient  we 
must  first  divide  each  term  of  the  equation  by  it ;  and  if  its 
coefficient  be  negative  we  must  change  the  signs  of  all  the ' 
terms. 


206  ELEMENTARY    ALGEBRA. 

Ex.  1.     Solve     -3x'  +  20x  +  6  =  0. 
Here  3:^;'- 20:^-5  =  0, 

and  x^—^^-x=^; 

therefore,  x'—^^-x+^^=^^; 

/,  o^^KlOiv'iis)^  the  roots  being  here  surd  quantities. 

Ex.2.      Solve     —. T\+-2 — T=-r- 

2(a;-l)     x^—1     4 

Here  we  first  clear  of  fractions  by  multiplying  by  ^x^^—l), 
which  is  the  least  common  multiple  of  the  denominators. 

Thus  2(a;  +  l)+12rr:cc'^-l. 

By  transposition,  a:'^— 22;=15; 

adding  1',  x'-2x +1  =  16 +  1  =  16; 

extracting  the  square  root,     a;—!  =  ±4; 
therefore,  a;=ldb4=5,  or  —3. 

Ex.3.     Solve       _+^-^^-=_^_. 

Multiplying  by  570,  which  is  the  least  common  multiple 
of  15  and  190,      ^ 

10 +  i?;  ^  ^ 

whence  190(^x-50)^^^^_^^ 

10  +  a;  ' 

and  190(3^-50)=:(210~40^)(10  +  ir); 

that  is,  570a;-9500=2100-190^-40a;'; 

therefore,  40a;''  +  7602:= 11600 ; 

or  a;'  +  19a;=290; 

addmg  (^— J  ,     a:''  +  19^+  (^yj  =290+-^-=--^  ; 


QUADRATIC    EQUATIOJS^S,  20T 

19         39 
extracting  the  square  root,  x  +—=  dz— ; 

whence  a;=— — dz— =10,  or  —29. 

^      ,      ^  ,         a;+3     i?:— 3      2:?;— 3 

Ex.4.     Solve     -— ^4- ^  = r- 

x+2     x—2       x—1 

Clearing  of  fractions, 

(x  +  d)  {x-2)  (^-l)  +  (^-3)  {x  +  2)  {x-1) 

=  (2x-d)  {x  +  2)  {x-2) ; 

that  is,    x'-'7x  +  6+x'-2x'-6x  +  Q=2x'~dx'Sx  +  12; 

or  2a;'-2a;'-122;4-12=2a;'-3:z;'^-8a;+12; 

therefore,  x^—4:X=0; 

adding  2^  *  x'-4.x  +  2':=4:; 

extracting  the  square  root,    a;— 2  =  ±2, 

whence  a;=2±2  =  4  or  0. 

Remark. — We  have  given  the  last  three  lines  in  order  to  complete 
the  solution  of  the  equation  in  the  same  manner  as  in  the  former 
examples ;  but  the  results  may  be  obtained  more  simply.  For  the 
equation  x^—Ax  =  ^  may  be  written  (a?— 4)a?=0;  and  in  this  form  it  is 
sufficiently  obvious  that  we  must  have  either  oj— 4=0,  or  x—0,  that  is, 
a^  =  4  or  0. 

The  student  will  observe  that  in  this  example  2a;^  is  found  on  both 
sides  of  the  equation,  after  we  have  cleared  of  fractions ;  accordingly, 
it  Ci,n  be  removed  by  subtraction,  and  so  the  equation  remains  a  quad- 
ratic equation. 

Examples — 52. 

6         1.  ■  X 

3.     ^x''-^x=^^{llx  +  lS).         4.     lla;^-9ri;-lli. 
5.     i{x'-3):=i{x-3),  6.     2r^Hl-:ll(.r+2). 


S:08  ELEMEJS'TARY    ALGEBRA. 


X 


1 


^      a;+33       4      9a;-6  ..      a;+3      4-3;  _ 


4 

^x-Q 

a; ' 

~      2     " 

4 

32 

12       _JL__?L  12        ^         a;4-l_13 

'  5-a;'^4-a;"a;  +  2'  a;+l         ri:    ~  6 ' 

231.  Sometimes,  on  completing  the  square,  the  second  side 
of  the  equation  becomes  0.  For  example,  take  the  equation 
ic^ — 14:^j  =  —  49.    This  giyes 

a;^-14a;  +  49=0  ;        .-.  {x-iy=0'y        .\  x=7. 

In  this  case  we  say  the  quadratic  equation  has  two  equal 
roots. 

232.  Solve    x'-^x  +  lZ^O, 

By  transposition,  x^  —Qx—  —  !^ ; 

adding  3',  a;'-6:z:+9  =  -13  +  9  =  -4. 

If  we  try  to  extract  the  square  root,  we  have 

cr-3=:±Vir4. 

In  this  case  the  quadratic  equation  has  no  real  root,  and 
this  is  expressed  by  saying  that  the  roots,  are  imaginary  or 
impossiUe, 

233.  An  equation  of  the  form  ax^-{-'bx-\-c=0,  or  ax^  ^hx 
=  — c  (where  a,  d,  and  c  are  any  quantities  whatever),  may 
be  solved  by  what  is  called  the  Hindoo  method,*  as  follows, 
without  diyiding  by  the  coefficient  of  x^.  Multiply  every 
term  hy  4a,  that  is,  4  times  the  coefficient  of  x^  a7id  add  V, 
that  is,  the  square  of  the  coefficient  of  x,  to  both  sides :  the  first 
side  will  be  a  "  complete  square."     Thus, 

4:aV  +  4.adx  +  b'=I?'-  4.ac. 

*  This  method  is  given  in  the  Bija  Ganita,  a  Hindoo  treatise  on 
Algebra. 


QUADRATIC    EQUATIONS.  200 

Extracting  the  square  root, 

.'.x=^{^d±>/¥^^^c).    (1) 

Ex.  1.    If  dx^  +  2:r=85,  find  x. 
Multiplying  by  4x3,  or  12,        3 6a;' +240; =1020; 
adding  2\  or  4,  36ic*+24a;  +  4=1024; 

extracting  root,  6a;  +  2=±32; 

B:r=:±32-2=:30,  or  -34; 
/.  x=z5,  or  — 5f. 

Ex.  2.     If  6x'  -dx^  2i=  0,  find  x. 
Transposing,  6x^ — 9x=  —  2  J ; 

multiplying  by  4x5,  or  20,     100:r'— 180^;=  -45 ; 
adding  9',  or  81,  100cc'-180a:+81  =  81-45=36; 

extracting  root,  lOx— 9  =  dz6; 

whence,  10a;=  9  ±  6  =  15,  or  3 ; 

15  3 

.•.:.=^,or-; 

=  4,  or-. 

The  student  will  find  it  well  to  apply  at  once  (by  memory) 
the  formula  (1)  above  obtained  for  x, 

Ex.  3.     (3:i;-2)  {l-x)=4:,  or  dx'-5x-{-6  =  0. 

Here  a:=:i(5±^^25-72)=l(5±^^^^^),   the  roots  being 
impossible. 

Or,  since  it  appears  that  the  equation  ax^  +  I)x-=  —  c  is  re- 
ducible to   the   simple   equation    2ax-\-b=:dt:^b''—4:ac,   the 

What  two  abbreviated  forms  of  Bolution  are  suggested  ? 

14 


^]0  ELEMENTAEY    ALGEBRA. 

student  may  readily  acquire  the  habit  of  writing  down  the 
simple  equation  from  any  proposed  equation  without  the  aid 
of  any  intermediate  steps. 

Ex.4.     Solve    3x'+6x=^2, 

62;  +  5  =  ±V25  +  12X42; 
that  is,  6a; + 5  =  ±23 ;        ,\x= 3,  or  —  4|. 


EXAMPLES- 


_f;? 


^        2x      2x-6     _.          _  2x  +  9    4.X-3     _     3x-16 

5a;        3x—2_^  4a;+7     5— a;_4a; 

•  ^T4~2a;-3~"'           '  19    "^3T^~"9'' 

x—1     ^-2_2^+13  ^+1     ^+^_2^+13 

•  x-\-l'^x  +  2~  x  +  1^'  *    a;-l'^a;-2~  ic+1  ' 

2a;-l     3^-l_5^--ll  14a:-9     a;^-3 

a;  +  l  "^  a;  +  2  ~  a;-!  *  *    ^~  8a;-3  "ic  +  l* 

9.    a'a;^-2a'i?;+a'-l  =  0.  10.     4a'a;=(a^-^^  +  a;)^ 

^^      X      a    x     h  ^^1111 

11-    r+-=r+-.  12.    r+i^^=-+: 


a      X    h     X  '    X     x  +  h     a     a+b' 

234.  If  r,  r'  represent  the  two  roots  of  x^+px+q=0, 
then  —p  =  r  +  r\  Sind  q=rXr'. 

YoTr=-lp-hViip'-q),    r'^-ip-VUp'-q) ; 
.-.  r  +  r'=—2y,  and  rXr'==iy~(iy— g)  =  ^. 

Hence,  luhen  any  quadratic  is  reduced  to  the  form 
x^'  +  px  +  q^O,  the  coefficient  of  2d  term,  ivith  sig?i  changed, 
=  su7n  of  roots,  and  the  3d  term  =  product  of  roots. 

To  what  are  the  sum  of  the  roots  and  the  product  of  the  roots  of  a  quadratic  equa- 
tion, respectively,  equal  ? 


QUADEATIC    EQUATIOXS.  211 

Thus,  in  (Ex.  1,  227)  the  equation,  when  expressed  in  this 
form,  is  x^  —  6x—7  =  0,  and  the  roots  are  there  found,  7  and 
—1;  and  here  -{■6  =  '7-\-{  —  l)  =^  sum  of  roots,  and  —7  = 
7X  (— 1)  =  product  of  roots. 

So,  also,  ax'^-\-hx  +  c=0,  expressed  in  this  form,  becomes 

x^-\ — x^ —  =  0;  .-. =  sum  of  roots,  —  =  product 

a        a  a  a      "- 

235.  If  T,  r'  he  the  roots  of  x^  ■\-px-\-q  —  ^,  then 

x^-\-px-\-q—(x—r^  (x—r'). 
For,  (236)  x^ -\rpx-\-q—x^ —(r  ^r')  x-^r  r\ 

=x^ —rx--r'x+r  r'  =  {x—r)  {x—r'). 

So,  also,  if  r,  r'  be  the  roots  of  ax'^+bx-{-c=0, 

b         c 
that  is,  of  x''-\ —  X-] — =0, 
a        a 

we  have  ax''  +  bx  +  c~a  (x^-i —  x  +  —]—a{x—r)  {x—r'). 

236.  Hence  we  may  form  a  quadratic  equation  with  any 
two  given  roots. 

Thus  with  roots  2  and  3  w^e  shall  have 

(^-2)  {x-^)=x''-bx-{-^=0. 

With  roots  —2  and  J,  we  have  {x  +  2)  (x—D—x' ■\-lx—i 
=0;  or  clearing  it  of  fractions,  4iz;^  +  7r?^— 2  =  0. 

If  one  of  the  roots  be  0,  the  corresponding  factor  will 
be  x—Oy  or  x. 

Thus  with  roots.  0  and  4,  we  have  x{x~4,)  —  0,  or  x^—A:X 
=  0.  (Compare  Ex.  4,  Art.  230.)  In  such  a  case,  then,  x  will 
occur  in  every  terr)i  of  the  equation,  and  may  be  struck  out 
of  each;  but  we  must  notice,  always,  that  when  we  thus 
strike  out  x  from  every  term  of  an  equation,  x—^  satisfies  the 
equation,  and  is  therefore  one  of  the  roots. 

A  quadratic  equation  may  be  formed  from  any  two  given  roots.     a;=0,  a  root. 


212  elementary  algebra. 

Examples — 54. 
Form  equations  with  the  following  roots  : 
1.     7  and  -3.  2.    f  and  -|.  3.     -6  and  -5. 

4.    2|  and  0.  5.     10  and  -10. 

6.    a-f—  and  a .  7.     — 1  +  n/2  and  — 1— \/2. 

a  a 

m 

XXXI.    Equations  which  may  be  solyed  like 
Quadratics. 

237.  There  are  many  equations  which  are  not  strictly 
quadratics,  but  which  may  be  solved  by  the  method  of  com- 
pleting the  square.    We  will  give  some  examples. 

238.  Ex.  1.    Solve    x'-W=^, 

Adding  (|)^  x'-W  +  {iy=^^-^=^i^; 

extracting  the  square  root,  a;^— |==b|; 

whence,  a;'=|d=|=:8  or  —1 ; 

extracting  the  cube  root,  a;  ==2  or  —1. 

This  method  applies,  evidently,  in  all  cases  where  the 
lowest  of  the  two  exponents  of  the  unknown  quantity  is  one- 
half  of  the  highest  exppnent. 

Ex.  2.     a;  +  4a;2=21;  required  x, 

a;+4a;^+4=r21+4=25; 

iz;i  +  2=:±5; 

a;*=:±5-2=3  or -7; 

:,x=^  or  49. 

What  other  equations  may  be  solved  like  quadratics  ? 


QUADRATIC     EQUATIONS.  213 

Ex.3,    a;"' +  ^""^=6;  required  ic. 

X  ^z= — - — =2  or  —6; 

.-.  x=l  or  f 
239.   Equations   may  be  proposed  containing   quadratic 
iTirds,  from  which,  by  performing  the  operations  of  trans- 
posing and  squaring,  once  or  oftener  (Art.  223),  we  obtain 
afiected  quadratic  equations. 

Ex.  1.    x+V6x-{-10  =  S  ;  to  find  x. 
By  transposition,  'v/52;+10=:8— :?:; 

squaring,  5a; +  10 =64— 16:?;+ a;'; 

x'-21x=-54c; 

441     441  225 

^«__21a;+— =— -54=— ; 

21 ^15 

21±15     ^_       .. 
a;= — ^r — =18  or  3. 

We  have  thus  found  two  values  of  x ;  but  on  trial  we  find 
that  18  does  not  satisfy  the  equation  if  we  suppose  that 
1/0:2;  + 10  represents  the  positive  square  root.  The  value  18 
satisfies  the  equation  x—  y  5.^  +  10=8. 

Ex.  2.     Solve  2x-V{x'-3x-3)  =  d, 

Transposing,  2x—9  =  V{x^—3x—3); 

squaring,  4:?;^--36a;  +  81=cc^— 3.^—3; 

transposing,  3^;'— 33.^+84=0; 

dividing  by  3,  '  x'-  llo;  +  28=0. 


214  ELEMEKTARY    ALGEBRA. 

By  solving  this  quadratic  we  shall  obtain  x^=^^l  or  4.  The 
value  7  satisfies  the  original  equation ;  the  value  4  belongs 
strictly  to  the  equation  2a;  +  \/(a;^  — 3a;— 3)  =  9. 

Ex.  3.     Solve      N/(aj  +  4)  +  v'(2:?;  +  6)3=N/{8^+9). 

Squaring,        aj4-4  +  2a;  +  6  +  2v/(:2;  +  4)v/(2a;4-6)  =  8a;  +  9 ; 

transposing,  2\/(:?;  +  4)\/(2a;  +  6)  =  52:— 1; 

squaring,  4(2;  +  4)  (2a;+6)  =  25^^— 10::i;  +  l; 

that  is,  8a;'  +  56a;  +  96  =  252;'— 10a;  +  l ; 

transposing,  17:?;^— 66:?;— 95  =  0. 

By  solving  this  quadratic  we  shall  obtain  cc=:o,  or  — f^ 
The  value  5  satisfies  the  original  equation ;  the  value  —  ^ 
belongs  strictly  to  the  equation 

v/(2^  +  6)-v/(:?;  +  4)=:x/(8:2;  +  9). 

240.  The  student  will  see  from  the  preceding  examples 
that  in  cases  in  which  we  have  to  square  in  order  to  reduce 
an  equation  to  the  ordinary  form,  we  cannot  be  certain,  with- 
out trial,  that  the  values  finally  obtained  for  the  unknown 
quantity  belong  strictly  to  the  original  equation. 

241.  Solve    x^ ■\-Zx-\-?>sf(x^ ^Zx-%):=^^. 
Subtracting  2  from  both  sides, 

a;'-f3:?;-2  +  3v/(^'  +  3a;— 2)=4. 

Thus  on  the  left-hand  side  we  have  two  expressions, 
namely,  \/(a^  -\-Zx—%),  and  x'^  +  ^x—2,  and  the  latter  is  the 
square  of  the  former ;  we  can  now  complete  the  square. 

Adding  (-|)^ 

i2;^  +  3^_2  +  3x/(^'  +  3a;-2)  +  (|y=4  +  |=^; 
extracting  the  square  root, 

N/(a;''  +  3a;-2)+|=zb|; 
therefore,         v/(:r'  +  3a;-2)  =  -f  ±f =1  or  -4. 


QUADRATIC    EQUATIONS.  215 

First  siii:)pos8  \/{x^  +  3x—2)  =  1 ; 

squaring  both  sides,  2;'^ 4-32'— 2  =  1. 

This  is  an  ordinary  quadratic  equation ;  by  solving  it  we 

T.  n    T4-  •                     ^       -3±^/21 
shall  obtain  x= . 

z 

l^ext,  suppose  V{x^-{-3x—2)  =  —4z. 

Squaring  both  sides,  r?;^+3:?;— 2  =  16. 

This  is  an  ordinary  quadratic  equation ;  by  solving  it  we 
shall  obtain  a; =3  or  —6. 

Thus  on  the  whole  we  have  four  values  for  x,  namely, 

-3±n/21 


3  or  —6,  or 


2 


o -4-/9-1 

But  we  shall  find,  on  trial,  that  only  the  values ■ 

2 

will  satisfy  the  given  equation 

x'+3x  +  3V{x'  +  3x-2)  =  6, 
but  the  values  3  or  —  6  satisfy  the  equation 

x''  +  dX'-3V{x'-{-dx-2)z=6. 

242.  The  method  pursued  in  the  example  in  the  last  article 
applies  whenever  an  expression  may  be  formed  which,  con- 
taining all  the  unknown  terms  outside  of  the  surd,  is  the 
same  as  the  surd  expressio?i,  or  is  a  rmilUple  of  it, 

2IB.  Equations  of  the  third  degree  are  sometimes  proposed 
in  which  it  is  intended  to  find  one  of  the  roots  by  inspection 
or  trial,  and  the  two  remaining  roots  by  solving  a  quadratic 
equation. 

x-{-A:      x—4:     d  +  x      9—x 


Ex.  1.     Solve 


X-\r4:     9  —  x       9-hx' 


Bring  the  fractions  on  each  side  of  the  equation  to  a  com- 
mon denominator.     Thus: 

What  method  of  solution  is  explained  iu  Art.  243? 


.216  ELEMEKTARY    ALGEBRA. 

16x  d6x 

that  IS, 


-16""81-cz;' 


Here  it  is  obyious  that  x=0  is  a  root  (Art.  236).  To  find 
the  other  roots  we  begin  by  dividing  both  sides  of  the  equa- 
tion by  ^x.     Thus : 

4__ 9_^ 

ic^  - 16  ~  81 -:?;'/ 

therefore,  4(81~:r^)=:9(a;'-16) ; 

/.  132;'=324  + 144=468; 
.-.  x'  =  36;  .-.   x=do6. 

Thus  there  are  three  roots  of  the  proposed  equation, 
namely,  0,  6,  —6. 

Ex.  2.     Solve  x'-7xa'  +  6a'=0. 

Here  it  is  obvious  that  a;=fl^  is  a  root.  We  may  write  the 
equation, 

x^—a''  =  '7a\x—a); 

and  to  find  the  other  roots  we  begin  by  dividing  hj  x—cu 

Thus,  x''-]-ax+a'=7a^. 

By  solving  this  quadratic  we  obtain  x=2a,  or  —3a.     Thus 
there  are  three  roots  of  the  proposed  equation,  namely  a,  2a 
3a. 

Examples — 55. 

1.  a;'-13.^'4-36=0.  2.     x-5Vx-U=0. 

3.  x-^V{x+^)=::Z  4.    x'+y/{x''  +  9)=21. 

5.  2v/(a;''-2a:+l)+:?;'=23  +  2a;. 

6.  x'-2x'+x'  =  36.      7.     9V{x'-9x-{-2S)  +  9x=x'i-S(>, 


QUADEATIC   EQUATIONS.  217 

9.  x'-4.x''--2V{x'-4.x'  +  4:)=31. 

10.  a;  +  2v/(:?;'  +  5:2;  +  2)  =  10. 

11.  ^x+^/{x''h'7x  +  6)  =  19, 

12.  v/(a;  +  9)  =  2v/.T-3.  13.     6  s/{l -x')+6x=7. 
14.  v/(3:c-3)+  N/(5a;~19)  =  v/(2aj  +  8). 

:?;+ \/(12<^''— ^)     a  +  l 


15 


X—  Vil^a^—x)     a- 


..>         1  1  1  1        ^ 

16.     ^  + -H -  + ^=0. 

x  +  7  ^— 1     cc  +  l     x—1 


x+V{2-x')^x-  V{2-x-')     ^ ' 


19 


x-^a    x—a     b  +  x    h—x 


x—a    x  +  a     h—x    b  +  x' 
20.     x'  +  3ax''=4:a\ 

XXXII.    Problems  which   lead  to  Quadratic  Equa- 
tions CONTAINING   OnE  UNKNOWN  QUANTITY. 

214.  In  the  solution  of  problems  depending  on  quadratic 
and  higher  equations  there  may  be  two  or  more  values  of  the- 
root,  and  these  values  may  be  real  quantities,  or  impossible. 
In  the  former  case,  we  must  consider  if  any  of  the  roots  are 
excluded  by  the  nature  of  the  question,  which  may  altogether 
reject  fractional,  or  negative^  or  surd  answers;  in  the  latter 
case,  we  conclude  that  the  solution  of  the  proposed  question 
IS  arithmetically  impossible. 

Prob.  1.  Find  two  numbers,  such  that  their  sum  is  13,  and 
their  product  is  42: 

What  is  said  of  the  different  roots  of  quadratic  equations  ? 

10 


21H  ELEMEJS^TARY    ALGEBllA. 

Let  X  be  one  of  the  numbers,  tben  Id—x  will  be  the  other ;  and  then, 
x{lS-x)=i2; 

x=7  or  6.        .-.  13-a;=:6  or  7. 

Thus  the  two  numbers  are  7  and  6.  Here,  although  the  quadratic 
equation  gives  two  values  of  i»,  yet  there  is  really  only  one  solution  of 
the  problem. 

Pkob.  2.  What  number,  when  added  to  30,  will  be  less 
than  its  square  by  12  ? 

Let  X  =  the  number ;  then 

whence  «  =  7or— 6. 

And  here  the  latter  root  would  be  excluded  if  we  required  only  posi- 
tive numbers. 

Pkob.  3.  A  person  bought  a  number  of  oxen  for  $600 :  if 
he  had  bought  3  more  for  the  same  money,  he  would  have 
paid  $10  less  for  each.     How  many  did  he  buy  ? 

Let  X  be  the  number  bought ;  then  the  price  actually  given  for  each 
was  — ,  and  therefore, 

600      600     ^^ 

x  +  S~  X  ' 

whence  a^=12  or  —15, 

which  latter  root  is  rejected  by  the  nature  of  the  problem. 

Prob.  4.  There  are  four  consecutive  numbers,  of  which,  if 
the  first  two  be  taken  for  the  digits  of  a  number,  that 
number  is  the  product  of  the  other  two. 

Let  X,  x  +  1,  x  +  2,  x  +  S,  be  the  four  numbers  required;  then 
10^  +  (aj  + 1)  =  the  number  whose  digits  are  x,  and  x  +  1. 

Therefore,  by  the  question,   (x •^2){x  +  S)  =  IO.2; -\-{x  +  l); 

or  x''  +  5x  +  e=zllx  +  l; 

whence  ic  =  5  or  1. 


QUADKATIC   EQUATIO^^S.  219 

Hence  the  numbers  required  are  5,  6,  7,  8,  or  1,  2,  3,  4,  both  of  which 
results  satisfy  the  conditions  of  the  problem. 

Prob.  5.  Find  two  numbers  whose  difference  is  10  and 
whose  product  is  one-third  of  the  square  of  their  sum. 

Let  X  =  the  smaller,  and  a;  +  10  =  the  greater ;  then, 

whence  2J=— 5±5\/^, 

which  values  are  impossihle.  And  the  solution  of  the  question  is 
arithmetically  impossible,  as  may  easily  be  shown,  since  it  calls  for 
two  numbers  whose  product  is  equal  to  the  sum  of  their  squares. 

245.  The  reason  why  results  are  sometimes  obtained,  as  in 
Prob.  3,  which  do  not  apply  to  the  problem  proposed,  seems 
to  be  that  the  algebraic  language  is  more  general  than  the 
ordinary  language  in  which  the  problem  is  stated ;  and  thus 
the  equation  which  expresses  the  conditions  of  the  problem 
will  also  apply  to  other  conditions.  It  will  be  a  profitable 
exercise  for  the  student,  when  it  is  possible,  by  suitable 
changes  in  the  statement  of  the  problem,  to  form  a  new 
problem,  corresponding  to  the  result  which  was  inapplicable 
to  the  original  problem.  Thus  in  Prob,  3  it  will  be  found 
that  "  15  "  oxen  is  the  answer  of  the  following  problem :  Find 
the  number  of  oxen  bought  for  $600,  when,  if  the  person 
had  bought  '^  fewer  oxen,  he  would  have  paid  $10  more  per 
head. 

Examples — 56. 

1.  Find  the  three  consecutive  numbers  whose  sum  is  equal 
to  the  product  of  the  first  two. 

2.  The  sum  of  two  numbers  is  60,  and  the  sum  of  their 
squares  is  1872  :  find  the  numbers. 


Why  are  some  of   the  results    ohtainefl   inapplicable? 


220  ELEMENTARY    ALGEBIIA. 

3.  The  difference  of  two  numbers  is  6,  and  their  product 
is  720 :  find  the  numbers. 

4.  Find  three  numbers,  such  that  the  second  shall  be 
two-thirds  of  the  first,  and  the  third  one-half  of  the  first, 
and  that  the  sum  of  the  squares  of  the  numbers  shall  be 
549. 

5.  Find  the  number  which  added  to  its  square  root  will 
make  210. 

6.  There  are  two  numbers,  one  of  which  is  f  of  the 
other,  and  the  difference  of  their  squares  is  81 :  find  them. 

7.  A  and  B  together  can  perform  a  piece  of  work  in  14| 
days,  and  A  alone  can  perform  it  in  12  days  less  than  B 
alone  :  find  the  time  in  which  ^. alone  can  perform  it. 

8.  In  a  certain  court  there  are  two  square  grass-plots,  a 
side  of  one  of  which  is  10  yards  longer  than  a  side  of  the 
other,  and  the  area  of  the  latter  is  -^^j  of  that  of  the  former : 
what  are  the  lengths  of  the  sides  ? 

9.  A  detachment  of  troops  was  arranged  in  a  column 
with  5  more  men  in  depth  than  in  front;  the  arrangement 
was  changed  so  as  to  increase  the  front  by  845  men ;  this 
left  the  column  5  men  deep :  find  the  number  of  men  in  the 
detachment. 

10.  There  is  a  rectangular  field,  whose  length  exceeds  its 
breadth  by  16  yards,  and  it  contains  960  square  yards:  find 
its  dimensions. 

11.  A  person  bought  a  certain  number  of  oxen  for  $1200, 
and  after  losing  3,  sold  the  rest  for  $40  a  head  more  than 
they  cost  him,  thus  gaining  $295  by  the  bargain:  what 
number  did  he  buy  ? 

12.  The  fore-wheel  of  a  carriage  makes  6  revolutions  more 
than  the  hind-wheel  in  going  120  yards ;  but  if  the  circum- 
ference of  each  were  increased  by  3  feet,  the  fore-whc^el  would 


SIMULTANEOUS   EQUATIONS.  221 

make  only  4  revolutions  more  than  the  hind  one  in  the  same 
space  :  what  is  the  circumference  of  each  ? 

13.  By  selling  a  horse  for  £24,  I  lose  as  much  per  cent,  as 
it  cost  me :  what  was  the  prime  cost  of  it  ? 

14.  Find  the  price  of  eggs  per  dozen,  when  two  less  in  24 
cents'  worth  raises  the  price  2  cents  per  dozen. 

15.  There  are  three  equal  vessels,  A,  B,  and  C ;  the  first 
contains  water,  the  second  brandy,  and  the  third  brandy 
and  water.  If  the  contents  of  B  and  C  be  put  together,  it 
is  found  that  the  mixture  is  nine  times  as  strong  as  if  the 
contents  of  A  and  (7 had  been  treated  in  like  manner:  find 
the  proportion  of  brandy  to  water  in  the  vessel  C, 

XXXIII.    Simultaneous    Equations    involving 

QUADEATICS. 

246.  We  shall  now  give  some  examples  of  simultaneous 
equations  which  may  be  solved  by  means  of  quadratics. 
There  are  three  cases  in  which  general  rules  can  be  given 
for' the  solution  of  these  simultaneous  equations  of  two  un- 
known quantities. 

217.  I.  When  one  of  the  equations  is  of  the  first  degree, 
and  the  other  is  of  the  second  degree : 

Eule. — From  tlie  equation  of  the  first  degree  find  the  ex- 
pression for  either  of  the  unhnoivn  quantities  in  terms  of  the 
other,  and  substitute  this  expression  in  the  equation  of  the 
second  degree. 

This  will  give  a  quadratic  equation  from  which  the  value' 
of  one  unknown  is  found. 

Example.     Given,    3:r4-4?/=18  ]  ,     ,,    -,  . 

^  ^     ^  ^  r  to  hnd  x  and  u, 

hx^  —  Zxy~  2  )  -^ 

Solution  of  Simultaneous  Equations  involving   quadratics.     Case  I. 


222  ELEME2!^TAIIY  ALGEBEA. 

From   the  first  equation,  y= — ;    substituting   this 

value  in  the  second  equation,  . 

whence,  20x''—54:X  +  9x''=8; 

that  is,  29x''-6^x=8. 

From  this  quadratic  we  shall  find  that 

x=2,ov-±; 

and  tlien  by  substituting  these  in  the  first  equation  we  find 
that 

2/=3,  or  --. 

218.  II.  When  the  two  equations  are  of  the  second  de- 
gree, and  all  those  terms  which  contain  x  and  y  are  homo- 
geneous, with  respect  to  these  quantities : 

EuLE. — Put  J=YX  in  both  equations;  obtain  by  division 
an  equation  in  wJiich  v  is  tJie  only  unknown;  v  being  deter- 
mined, X  and  J  may  then  be  found. 
Example.     Solve       2x''—xy=z6Q  \ 
2xy—y^=4.8  ) 
Putting  't^  —  vx  and  substituting  for  y, 

x\2-v)=^Q,  and  ^^(2^;--^;')=48; 
whence,  by  division, 

2v-v\    48  _  6  ^ 
2~v  ~56~  7  ' 

f  n 

w^hence,   v——,or,v=2.    The  latter  value  is  inapplicable; 
the  first  gives,  ^=±7,  ^—  ±  6. 

This  method  is  also  applicable  to  Case  I. 

Example.        x^  +  xy-\-y^=l 

2x-\ 


Case  n. 


SIMULTANEOUS   EQUATIOIs^S.  223 

V 

Here  puttiug  vx  for  y,    ' 

x\l  +  v  +  v'')=7     (1); 

x{2  +  dv)=S     (2). 

Therefore,  by  dividing  (1)  by  the  square  of  (2)  x"  disappears,  and  we 

*^^^'^'  (2  +  3^~64' 

whence,  ^=2,  or  18;  and,  from  (2), 

^i^-^'^)=^^      that  is,  i       ^=^'     and   i       2/=^^=^ ; 
or  aj(2  +  54)=8;  ( or  a,'=t;  ( or  y=vx=2^. 

249.  III.  When  each  of  the  two  equations  is  symmetrical 
with  respect  to  x  and  y,  put  u+v  for  x,  and  u— y  for  j, 

[Definition. — An  expression  is  said  to  be  symmetrical  with  respect 
to  X  and  y  when  these  quantities  are  similarly  involved  in  it.  Thus, 
each  of  the  expressions, 

x^-{-x^y'^  +  y^,         4xy  +  6x  +  5y—ly         2x'^—3x^y—dxy'^+2y*t 

is  symmetrical  with  respect  to  x  and  y.] 

Example.        x^+y^=18xy\      (1) 
x  +  y=12      [      (2). 

Put  u+v  for  X,  and  u—v  for  y  ;  • 
then  (1)  becomes  {u+vy+{u—vy  =  lS{u  +  v)  (u—v), 
or,  u'-{-3uv'=9(u'-v')  (3); 

and  (2)  becomes      {u  +  v)-{-{u—v)=12; 
whence,  u=6. 

Putting  this  for  u  in  (3),  • 

21Q-{-lSv'  =  9{d6-v'); 
whence,  t;  =  =h  2 ; 

.-.  x—u-{-v=6±:2=8oy4:; 
and,  ^='Z^— -^=6=^2=4  or  8. 

Case  III ;— Symmetrical  Equations. 


224  ELEMEKTAllY     ALGEBRA. 

250.  The  preceding  are  general  methods  for  the  solution 
of  equations  of  the  kind  referred  to^  and  will  sometimes  suc- 
ceed also  in  other  equations ;  yet  in  many  of  these  cases  a 
little  ingenuity  and  experience  will  often  suggest  steps  by 
which  the  roots  may  be  found  more  simply. 

Ex.  1.   Solve  3x'-2xy=16  )      (1) 
2x  +  3^=^:12  )      (2) 

Multiplying  (1)  by  3,  9x''-6xy=4t6, 

(2)  by  2x,        4a;'  +  6xy=2^; 


',  adding,  13cc'  =  45-f  24^;,  or  13.^^—240; =45,  whence  x=d  or 
~ly3^.     Equation  (2)  gives  y=^^  (12— 2:?;)  =2  or  4if. 

Ex.2.     Solve    x'+y'=26)      (1) 
2:?;^ =24  )       (2) 

Here  adding,        x^ +  2x2/ ^y"^ =4:9,  whence  a; +  ^=±7; 

subtracting,  x^  —  2xy-\-y''=:  1,  whence  iz;—^=:±l. 

If  x+y^  +  'l)  or,  if  x  +  y^+'l) 

and  x—y=-\-l )  and  x—y=  —  l  ) 


then  22^=8,  and  x=4:,  then  22;= 6,  and  x=3, 

also  2y=6,  and  ^=3;  also  2y=8,  and  ^=4. 

Similarly,  by  combining  the  equation  x-{-y=  —  7  with 
each  of  the  two  cc— ^=±1,  we  should  get  the  other  two 
pairs  of  roots, 

0;=— 4,  y=—dy  and  x=—3,  y=—4, 

Ex.  3.     Solve    x+y  =  b\      {1) 
x'  +  y'^^b)       (2) 

This  may  be  solved  by  the  method  of  Art.  249,  and  alstj 
as  follows : 

Bv  division,  — -^  =-^ ; 

x+y      5 

that  is,  x" -xy  +  ?/ =13.     (3) 


SIMULTANEOUS   EQUATIOIs^S.  225 

From  this  equation,  combined  witli  x+y=5,  we  can  f:nd 
X  and  2/  by  the  first  case,  or  we  may  complete  the  soluticn 
thus : 


x  +  y=5; 


(4) 


squaring,  x^  +  2xy  +  7/^=26; 

also,  (3)  x''-xy+if=13; 

therefore,  by  subtraction,    3:?;?/= 12; 
or,  xy=:4:; 

and,  4zxy—16,     (5) 

Subtracting  (5)  from  (4);  x''—2xt/  +  y^=9; 

extracting  the  square  root,  x—y=:iz3. 

We  haye  now  to  find  x  and  y  from  the  simple  equations 
x  +  y=:6,        x—y=^±:3. 

These  give  x=l  or  4,       y=^  or  1. 

Ex.4.     Solve    x'+xy+tf  =  19,        x'+xy  +  y*=U3, 

By  division,  __^:.__; 

that  is,  x'^—xy-\^y^=7. 

We  have  now  to  solve  the  equations 

x'^  +  xy  +  y'^=zl9,        x''—xy-hy^=7. 
By  addition  and  subtraction  we  obtain  successively 

a;'+/=z:13,        xy=6. 
Then  proceeding  as  in  Ex.  2,  we  shall  find 

x=dzd  or  d=2,        y=±2  or  =t:3. 

Examples — 57. 

1.    ^^{3x  +  6y)+i{^x-dy)  =  6U\  2.    x'  +  y'=2bl 

3.T^-h2/  =  179)  x-\-y=  1  ) 


22Q  ELEMEKTARY    ALGEBRA. 

3.      x'-\-y'=25)      4.    2{x-y)  =  ll\       5.   x'+xy=^m\ 
4^+3^=24)  xy=20)  x''-y'=.ll) 

6.  x-ij=2l-  '7.    x'=\^-y'-4:xyl 

16{x'-y'):=:^16xy)  x-y=2  ) 

8.      xy^{x-^)  (y  +  i)  I  9,   x  +y  =  6) 

xY^(x'  +  d)  (^^-4)  )  x'-Vf^T2  ) 

10.    3^^  +  2a;+^=485  )  11.    x  -y  —  \\ 

Zx=2y    \  x^-f^l9^) 

12.       a:'+^'  =  189)      13.    x-^y^a\       14.       xy=a^  \^ 
x^y^xy''=.V^^)  x'+y'  =  b')  x-^y=b  3 

15.    Vx  +  Vy=Sl  16.    x'-[-:.:i/=a') 

x+y=9)  y''  +  xy=lf) 

17.  l^  +  9{x  +  y)=2{X'^y)\        ^-{x-y)  =  {x-y)\ 

18.  x^—xy=a{x^-l)-\-h-^l,        xy—y'^—ay^-l. 


cc'     Z>  xy  x^      If'         '      xy 


21.   x^=^ax-\-'by,        y'^—ay-\-'bx. 

251.  We  shall  now  giye  some  problems,  to  be  solved  by 
equations  of  the  second  degree,  with  more  than  one  unknown 
quantity. 

Ex.  1.  The  sum  of  the  squares  of  the  digits  of  a  number 
of  two  places  is  25,  and  the  product  of  the  digits  is  12 :  find 
the  number. 

Let  X,  y  be  the  digits,  so  that  the  number  will  be  \^x^-y\ 
then  2:^+^^=25,  and  xy—\2,  from  which  equations  we  get 
a; =3,  y  —  ^^  or  a: =4,  ^=3,  and  the  number  will  be  34  or  43. 

In  this  case  both  the  roots  give  solutions. 

Ex.  2.  Find  two  numbers,  such  that  their  sum,  their 
product,  and  the  difference  of  their  squares  may  be  all  equal. 


SIMULTAISTEOUS   EQUATIONS.  22? 

Here  assume  x+y  and  x—yioT  the  two  numbers;  [this 
step  should  be  noticed,  as  it  simplifies  much  the  solution  ol' 
problems  of  this  kind :]  then  their  sum  =  2x,  their  product 
—x^^y"^,  and  the  difference  of  their  squares  =42;^; 

.-.  (1)  2x^4.xy,    (2)  2x=x''~y''] 

from  (1),  y=i-;  from  (2),  2x=x^—^; 
whence,  2;=  |-(2=fc  v/5) ; 

and,  .\  x  +  y=i  {3±V6),   x—y=i{l±:V6), 

the  numbers  required. 

Ex.  3.  A  man  sets  out  from  the  foot  of  a  mountain  to 
walk  to  its  summit.  His  rate  of  walking  during  the  second 
hal£  of  the  distance  is  half  a  mile  per  hour  less  than  his  rate 
during  the  first  half,  and  he  reaches  the  summit  in  5^  hours. 
He  descends  in  3|  hours,  by  walking  at  a  uniform  rate,  which 
is  one  mile  per  hour  more  than  his  rate  during  the  first  half 
of  the  ascent:  find  the  distance  to  the  summit,  and  his 
rates  of  walking. 

Let  2x  denote  the  number  of  miles  to  the  summit,  and 
suppose  that  during  the  first  half  of  the  ascent  the  man 

X 

walked  y  miles  per  hour.    Then  he  took  —  hours  for  the  first 

X 

half  of  the  ascent  and  — —  hours  for  the  second. 
^-2 
Therefore,  ^-|—L-=5i     (1). 

Similarly,  ^==3f     (2). 

From  (2),  2^=^(f/  +  l); 


15 
therefore,  x=—{y-\-l). 


4 

~  8' 


228  ELEMEJS'TAllY    ALGEBEA. 

From  (1),  x(2y-j^=~tj(^y-jy 

Therefore,  by  substitution, 

whence,  16{y  +  l)  {4:y--l)—Uy{2y—l); 

and,  28^'— 8% +  15  =  0. 

5 

From   this   quadratic    equation   we   obtain   y—3,   or  -^, 

/Co 

5 

The  value  -^   is  inapplicable,  because,  by  supposition,  y  is 

1  15 

greater  than  — .     Therefore,  y  =  3;  and  then  x=:-^,  so  that 

the  whole  distance  to  the  summit  is  15  miles. 

Examples — 58. 

1.  The  sum  of  the  squares  of  two  numbers  is  170,  and  the 
difference  of  their  squares  is  72 :  find  the  numbers. 

2.  The  product  of  two  numbers  is  192,  and  the  sum  of 
their  squares  is  640 :  find  the  numbers. 

3.  The  product  of  two  numbers  is  60  times  their  differ- 
ence, and  the  sum  of  their  squares  is  244 :  find  the  numbers. 

4.  Find  two  numbers,  such  that  twice  the  first,  w^ith  three 
times  the  second,  may  make  60,  and  twice  the  square  of  the 
first,  with,  three  times  the  square  of  the  second,  may  make  840. 

5.  Find  two  numbers,  such  that  their  difference  multiplied 
into  the  difference  of  their  squares  shall  make  32,  and  their 
sum  multiplied  by  the  sum  of  their  squares  shall  make  272. 

6.  Find  two  numbers,  such  that  their  difference  added  to 
the  difference  of  their  squares  may  make  14,  and  their  sum 
added  to  the  sum  of  their  squares  may  make  26, 


EXAMPLES.  229 

7.  Find  two  numbers,  such  that  their  product  is  equal  to 
their  sum,  and  their  sum  added  to  tlje  sum  of  their  squares 
equal  to  12. 

8.  The  difference  of  two  numbers  is  3,  and  the  difference 
of  their  cubes  is  279  :  find  the  numbers. 

9.  A  man  has  to  trayel  a  certain  distance,  and  when  he  has 
travelled  40  miles  he  increases  his  speed  2  miles  per  Hour. 
If  he  had  trayelled  with  his  increased  speed  during  the 
whole  of  his  journey,  he  w^ould  have  arrived  at  his  des 
tination  40  minutes  earlier,  but  if  he  had  continued  at  his 
original  speed  he  would  have  arrived  20  minutes  later :  find 
the  whole  distance  he  had  to  travel. 

10.  A  number  consisting  of  two  digits  has  one  decimal 
place ;  the  difference  of  the  squares  of  the  digits  is  20,  and 
if  the  digits  be  reversed,  the  sum  of  the  two  numbers  is  11  * 
find  the  number. 

11.  A  person  buys  a  quantity  of  wheat,  which  he  sells  so 
as  to  gain  5  per  cent,  on  his  outlay,  and  thus  clears  £16.  If 
he  had  sold  it  at  a  gain  of  5  shillings  per  quarter,  he  would 
have  cleared  as  many  pounds  as  each  quarter  cost'  him  shil- 
lings: find  how  many  quarters  he  bought,  and  what  each 
quarter  cost. 

12.  Two  trains  start  at  the  same  time  from  tw^o  towns, 
and  each  proceeds  at  a  uniform  rate  toward  the  other  town. 
When  they  meet  it  is  found  that  one  train  has  run  108  miles 
more  than  the  other,  and  that  if  they  continue  to  run  at  the 
same  rate  they  will  finish  the  journey  in  9  and  16  hours  re- 
spectively :  find  the  distance  between  the  towns,  and  the  rates 
of  the  trains. 

13.  A  and  B  take  shares  in  a  concern  to  the  amount,  alto- 
gether, of  $2500;  they  sell  out  at  par — A  at  the  end  of  2 
years,  B  at  the  end  of  8  years — and  each  receives,  in  capital 
and  profit,  $1485:  how  much  did  each  embark? 


230  ELEMENTARY    ALGEBRA. 

14.  Find  three  numbers,  such  that  if  the  first  be  multi- 
plied  by  the  sum  of  tha  second  and  third,  the  second  by  the 
sum  of  the  first  and  third,  and  the  third  by  the  sum  of  the 
first  and  second,  the  products  shall  be  26,  50,  and  56. 


XXXIV.    Eatio. 

252.  Ratio  is  the  relation  which  one  quantity  bears  to  an- 
other with  respect  to  magnitude,  the  comparison  being  made 
by  considering  what  multiple,  part  or  parts,  the  first  is  of 
the  second ;  or,  in  other  words,  what  fraction  the  first  is  of 
the  second.  Thus,  if  one  quantity  be  two-thirds  of  another 
quantity,  the  former  is  said  to  be  to  the  latter  in  the  ratio  of 
2  to  3,  for  if  both  be  divided  into  respectively  equal  parts, 
the  former  will  contain  two,  and  the  latter  three  of  these 
equal  parts.  And  thus  the  ratio  of  2  to  3  and. the  fraction 
f,  express  the  same  idea ;  for  f  indicates  that  unity  has  been 
divided  into  3  equal  parts,  and  two  of  them  are  taken. 

253.  The  ratio,  then,  of  one  quantity  to  another,  is  repre- 
sented by  the  fraction  obtained  by  dividing  the  former  by 
the  latter.     Thus,  the  ratio  of  6  to  3  is  |,  or  2 ;  that  oi  aioh 

is  7-;  that  of  15  to  40  is  77:,  or  -;  that  of  4a  to  6Z>  is  ^,  or 

2a 

^y.     Of  course  the  two  quantities  compared  must  be  of  the 

same  kind,  or  one  could  not  be  a  fraction  of  the  other.  (See 
Venable's  Arithmetic,  Art.  171.) 

254.  The  ratio  oi  aio  h  is  expressed,  either  by  two  points 
placed  between  the  quantities,  as  a :  5,  or  for  shortness,  by 

its  measure,  -r-.     The  first  of  the  quantities,  a:!),  is  called 

the  antecedent  term  of  the  ratio,  and  the  latter  the  consequent. 


Ratio.    Antecedent  and  Coneequent. 


EATIO.  J>31 

255.  A  ratio  is  said  to  be  of  greater  or  less  ijiequaliiy  ac- 
cording as  the  antecedent  is  greater  or  less  than  the  conse- 
quent. 

25G.  If  the  antecedents  of  any  ratios  are  multiplied  to- 
gether, and  also  the  consequents,  a  new  ratio  is  obtained, 
which  is  said  to  be  compounded  of  the  former  ratios.  Thus, 
the  ratio,  ac :  hd^  is  compounded  of  the  two  ratios,  a :  1), 
and  c :  d. 

When  the  ratio  a :  Z>  is  compounded  with  itself,  the  re- 
sulting ratio  is  a^  '.h^\  this  ratio  is  called  the  duplicate  ratio 
of  (2 :  Z> ;  and  the  ratio  a^ :  V^  is  called  the  triplicate  ratio 
of  a:h, 

257.  Problems  upon  ratios  are  solved  by  representing  them 
by  their  corresponding  fractions,  and  then  treating  these 
fractions  by  the  ordinary  rules.     Thus, 

If  the  terms  of  a  ratio  he  riiultiplied  or  divided  hy  the  same 
quantity,  the  ratio  is  not  altered, 

^  a     ma 

'  h     mo 

Thus  ratios  are  com/pared  with  one  another  by  reducing  the 
fractions  which  measure  these  ratios  to  common  denomina- 
tors, and  comparing  the  numerators;  and  they  are  com- 
pounded by  multiplying  together  the  fractions  which  meas- 
ure them.  Thus,  also,  a  ratio  may  be  reduced  to  its  lowest 
terms  by  dividing  the  numerator  and  denominator  of  its 
fraction  by  their  g.c.d. 

Ex.  1.   Compare  the  ratios  5  :  7  and  4  :  9. 

■^iis.  14,   H;  whence  5  :  7  >  4 :  9. 
Ex.  2.   Find  the  ratio  of  | :  f .     Ans.  4  -^  |-=|  X  |=:||. 
Ex.  3.   What  is  the  ratio  compounded  of  2 : 3,  6:7,  14 :  15  ? 
•       • Ans.  f  X  f  X  ii=TV  or  8  :  15. 

Ratio  of  greater  or  less  inequality.     Componnd,  duplicate,  and  triplicate  ra« 
t7X)s.    Solution  of  problems  upon  ratios. 


232  ELEME^^TARY    ALGEBRA. 

Ex.  4.    Reduce  to  its  lowest  terms,  a''—x^ :  a^  ■{■2ax-\'X^. 

.         (a—x)  (a  +  x)     a—x 
Ans.  7 — 7—xr~-. — \= — ; — ?  ^^  a—x:a+x, 
{a  +  x)  [a+x)     a+x 

258.  If  to  botli  terms  of  the  ratio,  a  :  b,  the  quantity  x  bo 
added,  that  ratio  will  be  increas'ed  or  diminished  according  as 
a  is  less  or  greater  than  b. 


For, 

a   .    ^ 

^  a-\-x 
b  +  x' 

if 

ab  +  ax  . 
b{b^x)        ^ 

ab-hbx 

"'  <  b{b+xy 

that  is. 

if 

ab-\-ax  > 

or  <  ab+bx; 

that  is, 

if 

ax  > 

or  <  bx; 

or  if 

a  > 

or  <  b; 

which  shows  the  truth  of  the  proposition. 


Examples — 59. 

1.  Compare  the  ratios  3  :  4  and  4:5;  13:14  and  23  :  24 ; 
3:7,  7  :  11,  and  11 :  15. 

2.  Of  a-\-b:a  —  b   and   a"^ +  b'^  :a^—b^,  which  is  greater, 
supposing  a>b? 

3.  What  is  the  ratio  b  inches  to  c  yards  ? 

4.  Find  the  ratio  compounded  of  3  :  5, 10  :  21,  and  14  :  15 ; 
of  7  :  9,  102  :  105,  and  15  :  17. 

a  ~\~  ax  -f-  x^ 

5.  Find  the  ratio  compounded  of  -^ ^ and 

.  a  —ax  +  ax^—x 

a^—ax-\-x^ 


a+x 

6.  Compound 

x'-9x+20  :x^-6x  and  x'-13x+A2  :x^~5x. 


RATIO.  233 

7.  Compound  the  ratios  a-\-h  :  a—d,  a^-{-b'^  :  {a-hby, 
{a'-by:a'-I)\ 

8.  "What  is  the  ratio  compounded  of  the  duplicate  ratio 
of  a+b :  a—b,  and  the  difference  of  the  duplicate  ratios  of 
a  :  a  and  a  :  b,  supposing  a>b? 

9.  What  quantity  must  be  added  to  each  term  of  the 
ratio  a  :  b,  that  it  may  become  equal  to  the  ratio  c:d? 

10.  Show  that  a—b:a  +  b^a^—b^:a^  +  b'^,  according  as 
a:  b  is  Si  ratio  of  less  or  greater  inequality. 

11.  Find  two  numbers  in  the  ratio  of  3  to  4=,  such  that 
their  sum  has  to  the  sum  of  their  squares  the  ratio  of  7 
to  50. 

12.  Find  two  numbers  in  the  ratio  of  5  to  6,  such  that 
their  sum  has  to  the  difference  of  their  squares  the  ratio  of  1 
to  7. 

13.  Find  x  so  that  the  ratio  x  :  1  may  be  the  duplicate  of 
the  ratio  Six, 

14.  Find  x  so  that  the  ratio  a  —x  :b--x  may  be  the  dupli- 
cate of  the  ratio  a  :  b. 

XXXY.    Proportion. 

259.  When  two  ratios  are  equal,  the  four  quantities  com- 
posing them  are  said  to  be  proportional  to  one  another ;  thus, 

(t  c 
a\b~c\d\  that  is,  \i  ——~,  then  a,  b,  c,  d,  are  proportion- 
als. Thus,  four  quantities  are  proportional  when  the  first  is 
the  same  multiple, part  or  parts  of  the  second,  as  the  third 
is  of  the  fourth.  This  is  expressed  by  saying  a  is  jfo  b  as  q 
is  to  d,  and  denoted  thus,  a\b\\c\d\  or  thus,  a\b  =  c:d\ 

or  thus,  —  —  —, 
b      d 

Proportion. 


234  ELEMENTARY    ALGEBRA. 

The  first  and  last  terms  in  a  proportion  are  called  the  ex- 
tremes, the  other  two  the  means. 

Problems  on  proportions,  like  those  on  ratios,  are  solved 
by  the  use  of  fractions. 

260.  (1.)  When  four  qua7iUties  are  proportionals,  the 
Xyroduct  of  the  extremes  is  equal  to  the  product  of  the  means, 

a      c 
For  if  —  =  — ,  then  ad—dc, 
h      d 

(2.)  Hence,  if  three  terms  of  a  proportion  are  given,  we 
can  find  the  other  from  the  equation  ad—lc.    Thus 

Ic       .ad  ad        ^    he 

"=!'      ^=T'      'S^      ^=a- 

(3.)  If  a :  1=^1 :  d,  we  have  ad—I'' ;  that  is,  if  the  first 
be  to  the  second  as  the  second  is  to  the  third,  the  product  of 
the  extremes  is  equal  to  the  square  of  the  mean. 

In  this  case  a,  l,  d,  are  said  to  be  in  continued  proportion, 

261.  If  the  product  of  two  quantities  he  equal  to  that  of 
tivo  others,  the* four  are  proportionals,  the  factors  of  either 
product  heing  the  extremes,  and  of  the  other  the  means. 

For,  let  ad— he, 

dividing  by  M,  ,         y=^7^ 

or  a\l—c\d, 

262.  \i  a\'b—c\d,  and  c\d=e:f,  then  a\h—e:f, 

_,  a      c        ^   c      e 

For  _^_and^=-^; 

therefore,  -7-=-^,  or  a:  J=e: /l  ' 

263.  li  a:'b=c:d,  and  e:f—g:h,  then  ae:hf=cg:dh. 

Extremes.    Means.    Demonstrate   Art.  260    (1),  (2),  and  (3).    Demonstrate  Art. 
261;    Art.   262;    Art.   263. 


PROPOitTiois".  235 

1-1  a      c         ^  e      a 

For  _=_and-^=^; 

^  ae      eg  t.^  ^j-l 

This  is  called  comjpounding  the  two  proportions.  And  so 
we  may  compound  any  number  of  such  proportions.  Thus, 
if  a\l  —  c  :  d,  a" :  If—c" :  d%  &c. 

2M.  If  four  quantities  le  proportionals,  they  are  propor- 
tionals when  talcen  inversely.  That  is,  if  a:h=c:d,  then 
'b:a=d\c. 

For  (Art.  134,  i),  if      |  =  |,     l-^f=l--|; 

that  is,  —  =  — ,  or  Z>:  a=:<^:  c. 

a      c 

265.  If  four  quantities  le  proportionals,  they  are  propor- 
tionals when  talcen  alternately.  That  is,  if  a:d=c'.d,  then 
a :  c=h:  d. 

For  (Art.  134,  u),  since  -r=-j,    -yX  — ^-yX  — ; 
0      d       0      c      d      G 

that  is,  '  — =-7»  or  a  :  c=d  :  d. 

c      a 

266.  If  four  quantities  are  proportionals,  the  first  together 
loitli  the  second  is  to  the  second,  as  the  third  together  with  the 
fourth  is  to  the  fourth, 

therefore  (Art.  134,  iii),  —j—— ,  or  a-\-'b  :  d=c  +  d:c, 

0  c 

267.  Also,  the  excess  of  the  first  above  the  second  is  to  the 
second  as  the  excess  of  the  third  above  the  fourth  is  to  the 
fourth. 

Art.  264;    Ai't.  265;    Art.  266;    Art,  267. 


236  ELEMENTARY    ALGEBRA. 


(X        c 
For  -7-  — -7;    therefore    (Art.  134,  iv),   by   subtracting   1 


from  each  of  these  equals, 

a—1)     c—d  77  -,    , 

—7 — = — 7-,  OY  a—o:  o~c—d\d. 
0  d 

2G8.  We  have  also  (134,  v), 

a^l)      d     c:^d     d  adtzh     c^d 


hade  a  c 

or  a±Z> :  a—G::^d :  c,  which,  by  inversion  (264),  gives 
a'.a-±^l)=c\c±d. 

269.  When  four  quantities  are  proportionals,  the  sum  of 
the  first  and  second  is  to  their  difference  as  the  sum  of  the 
third  and  fourth  is  to  their  difference. 

For  (Arts.  266  and  267), 

ct-\-l)      c-\-d        -,  a—h     c—d 


therefore, 
that  is, 


d    ^    d  '  l    ~    d 

a+d  ^  a—h_ci-d     c—d^ 

a—h~c—d'^ 
or    a-\-'h\a'-'b^c-^d\c—d. 


270.  If  four  quantities  form  a  proportion,  we  may  der.ye 
from  them  many  other  proportions*,  all  equally  true. 

Thus,  if  —  =— ,  then  — 7=-^  or  ma  :  m'h~c\  d, 
0      d  mo     d 

Similarly,    ma  :  h — mc  \  d,    a\mJ)=c:md,    a:d  —  mc\  md ; 

and  in  like  manner 

ad  ,  h  d    ^ 

—  :  —  =c:dy     a:  —  =c:  -  ,  &c. 
mm  mm 

Art.  268;    Art.  269  ;  Art.  270, 


PK0P0RTI02s\  237 

That  is^  either  the  Jirst  or  fourth  terms  of  any  proi)ortion 
may  be  multiplied  or  divided  by  any  quantity,  provided  that 
either  the  second  or  tliird  be  multiplied  or  divided  by  the 
same. 

Ct         G 

271.  As'ain ;       if  a:I?=c:d,  then  — =  — ; 

°  0      d 

m     a     m      c         ma    mc 
and  — X-7  =  — X-T?  or  —.=—.; 

n     0      n     d         no     nd 

or  ma :  nh  —  mc  :  na. 
Then,  by  the  preceding  Articles,  or  by  Art.  134, 

ma±7ib    mc±;?f/^ 
7)ia     '~     mc      ^ 

madcnl)    fnc^hnd 

whence,  also, = ; 

a  c 

or  ma dtznb:  a  =  mc±nd:  c, 

A2:am(Art.  269); 7= •. 

ma— no     m  0—714 

or  ma-{-nI):ma—nd=mc  +  nd:mc—nd, 

272.  (1)  In  like  manner,  if  a:b=c:  d—e  :/,  &c.,  by  which 

a      c      6 
it  is  meant  a  :  h=c  :  d,  or  a  :  h=^e  :  f,  &c.,  so  that  —=1—=--:, 

-^  h      d      f 

&c.     Then  r^ :  Z>=a  +  c+6  :  l-^-d^-f,  &c. 

For,  let  — =ri?^=— =— ;  i\\Qn  a—hx,     c—dx,     e=fx\ 
h  d     f  ^         J    ^ 

therefore,         a-^-c-^-e—lx-^dx  -\-fx  —{h  +  d  4-/ )  x, 

.-.  a;  or  — =^-— -— -  ,  or  a:  o=a-\-c+e:d  +  d-}-f, 
0     o-i-d-\-f 

That  is,  if  there  be  a7iy  7iumber  of  quantities  in  projjortioii, 
as  one  antecedent  is  to  its  consequent,  so  is  the  sum  of  all  the 
antecedents  to  the  sum  of  all  the  co7iseqttents. 

Art.  271;    Art.  272  (1),  (2),  (3). 


238  ELEMEKTARY    ALGEBRA. 

(I  C 

Again  (2) ;  the  equations  above  deriyed  from  — =a;=-y 
=-^5  &c.,  give  ma^=^mbx^   nc=7idx,  pe=pfx,  &c. 

/.  7na-\-nc  +2^c  —  (in h-\-nd -\-pf  ^x\ 

.,       „  a     ma  +  nc+pe 

therefore,       .         x  ot -:r-=^ 7-^; 

•  0     mo-\-na+pf 


a  _c 


So  also  (3),  since  -.=-^-=:-,  &c.,       r^=-  =  -; 


Since  d=-  (Art.  260),  this  is  ^=i. 


, T       r,  a  _a  -\-c  -{-e   _ ma  +  nc  -^pe 

merelore,  -_---— --^_^^^^_^^^^^^^ ; 

and  so  on  for  any  number  of  terms  and  any  like  powers. 
Ex.  1.  Find  a  fourth  proportional  to  \,  J,  and  \. 

a  ^  "  '     I- 

Ex.  2.  Find  a  mean  proportional  to  2  and  8. 

Since  Vz=iaG  (Art.  260),  this  is  n/(2x8)=  v/16  =  4. 

Ex.  3.  \i  a\'b—c\d^  express  (a^d)  —  {b-\-c)  in  terms  of  a, 
Z>,  c  only. 

Here  {a^-d)-{l^c)  =  (a^^-^--{M-c) 

^a^—ab—ac-\-'bc_{a—b)  {a—c) 
~^  a  "'  a  ' 

Examples— 61. 

1.  Find  a  fourth  proportional  to  3,  5,  6 ;   to  12,  5,  10 ' 
to  h  h  h 

2.  Find  a  third  proportional  to  4,  6 ;  to  2,  3 ;  to  |,  \, 


PROPOETIO^^  2o9 

3.  Find  a  mean  proportional  to  4,  9 ;  to  4,  |-| ;  to  1|,  l-^^g-. 

4.  1^  a '.!):'.  h'.c,  then,  a^ -^If  \a  +  c:\  a^—W  :  a  —  c. 

5.  If  |  =  |,  show  that  (a^l)  (0  +  (^)r=^((;  +  ^)'^r:r|^(a  +  ^)% 

6.  If  a  :  Z> : :  c  :  ^,  and  m:n\\p:q, 

then  ma-\-7id  :  ma—nb  :  :pc  +  qd  :pc—qd, 

7.  If  a\h\\l)\c,  then  a^— Z^'^ :  «^ : :  Z^"^  — c'^ :  c. 

8.  li  a\l)\:c\d:\e:f,  then  a— e  :  Z>— /: :  c  :  d 

9.  If  ^  :  Z> : :  Z> :  (?, 

then  mo^—'nF  \ma—nc\  ipa^-i-qH^  :pa-{-qc. 

Solve  the  equations, 

10.  -Vx^  y/h'^  \/X'-\/'b=a:'b, 

11.  2;  +  a:  2.?;— ^:=3a;  +  ^  :4a;— fl^. 

12.  a;+y  +  l::r+^+2=:6:7 

y +  2:^; :  ;z/-2a;=12a;  +  62/— 3  :  62/'-12a;-l. 

13.  xi^'l—y.^—^-.x—y, 

14.  What  number  is  that  to  which  if  1,  5,  and  13  be  sev- 
erally added,  the  first  sum  shall  be  to  the  second  as  the  sec- 
ond to  the  third  ? 

15.  Find  two  numbers  in  the  ratio  of  2^ :  2,  such  that, 
when  diminished  each  by  5,  they  shall  be  in  that  of  l-J- :  1. 

16.  A  railway  passenger  observes  that  a  train  passes  him, 
moving  in  the  opposite  direction,  in  2'^,  whereas,  if  it  had 
been  moving  in  the  same  direction  with  him,  it  would  have 
passed  him  in  30" :  compare  the  rates  of  the  two  trains. 

17.  A  quantity  of  milk  is  increased  by  watering  in  the 
ratio  of  4  : 5,  and  then  three  gallons  are  sold ;  the  rest  being 
mixed  with  three  quarts  of  water,  is  increased  in  the  ratio 
of  6:7:  how  many  gallons  of  milk  were  there  at  first  ? 


240  ELEMEA^TAEY    ALGEBRA. 

XXXVI.    Arithmetical  Progression. 

273.  Quantities  are  said  to  form  a  Series  when  they  ^7-0- 
ceed  by  a  laio,  i.  e.,  when  any  one  quantity  may  be  obtained 
from  those  which  precede  it  by  a  rule,  which  is  the  Imv  of 
the  series. 

274.  Quantities  are  said  to  form  an  Aritlimetical  Series,  or 
to  be  in  Arithmetical  Progression,  when  they  proceed  by  a 
common  difference. 

Thus,  the  following  series  are  in  a.p.  : 

1,  3,  5,  7,  9, 

12,  8,  4,  0,  -4,  ...  . 

a,  a-\-d,  a  +  2d,  a  +  3d,  .... 

In  the  first  and  third  the  quantities  increase  as  the  series 
proceeds ;  in  the  second,  they  decrease ;  the  common  differ- 
ences being  2,  —4,  and  d,  respectively,  which  are  found  by 
subtracti7ig  any  term  from  tlie  term  folloiving ;  therefor.?, 
when  the  progression  is  a  decreasing  one,  the  common  differ- 
ence is  negative. 

275.  Given  a,  the  first  term,  and  d,  the  common  difference 
of  an  Arithmetical  Series,  to  find  1  the  n*^  term,  and  S  the 
sum  of  n  ter^ns. 

Since  a  is  the  first  term,  and  d  the  common  difference, 
the  second  term  is  a  +  d',  the  third  term  is  a-\-2d',  the 
fourth  term  is  a  +  3d;  and  so  on,  where  the  coefiicient  of  d 
is  less  by  one  than  the  number  of  the  term.  So  in  the  71^^ 
term  we  shall  have  {7i—l)d;  therefore,  the  71"'  term 

l=a+{n-l)d.     (1) 

Again,  the  sum  of  the  terms, 

S=a'h{a+d)  +  {ai'  2d)  +  &c.,  -f  {I-  2d)  +  {l-d)+l; 

Series.  Arithmetical  Series,  or  Progression.  Common  difference,  how  found; 
when  ne^tive.    How  to  find  the  last  term,  and   the  sum  of  the  scricB. 


AKITHMETICAL   PROGKESSIOK^.  241 

and  by  writing  the  series  in  the  reverse  order,  we  have  also 
8=l-{-{l-d)-{-{l-M)-\-kQ.  +  {a+M)  +  {a^cl)-\-a. 
Therefore,  by  addition, 

2/S'=(«  +  Z)  +  (^+?)  +  (a+Z)  &c.,  to  n  terms; 
.-.  ^8^  {a +  1)71', 

and  since  l=a-\'{n—l)d, 

we  have  also,  S=\^a-^{n—l)cT\—.     (3) 

The  equation  (2)  furnishes  the  following  rule : 

The  sum  of  any  nmnber  of  terms  in  A.  p.  ^5  equal  to  the 
product  of  the  numher  of  terms  into  half  the  sum  of  the  first 
and  last  terms. 

Ex.  1.  Find  the  sum  of  20  terms  of  the  series  1,  2,  3,  4. 

Here  a=l,  d—l,  ^^=20;  using  formula  (3), 

;S^=r[2  +  (20-l)lp/,  or  =(2  +  19)-2/=21XlO=210. 

Ex.  2.  Find  the  9th  term,  and  the  sum  of  9  terms  of  7, 
5i,  4,  &c.  -^ 

Here  ^=7,  ^=— |,  7^— 9; 

.-.  Z==7+(9-l)x-|=7-8x|=~5; 

and  ^=1(7-5)^9. 

Ex.  3.  Find  the  13th  term  of  the  series  -48,  -44,  —40,. 
&c. 
Here  a=  -48,  ^==4,  n=ld', 

.-.  Z==-48  + (13-1)4=0. 

Ex.  4.   Find  the  sum  of  seven  terms  of  i+i  +  i,  &c. 

Formulas  (1),  (2),  (3). 
11 


242  ELEMEiifTARY    ALGEBRA. 

Here  a  —  ^,  d=—^,  71=7;    here,  as  in  Ex.  1,  we  are  not 
required  to  find  I;  ,\  using  formula  (3), 

^zz.(l  +  6X-i)l-(l-l)J=0. 
'  The  series  continued  to  seven  terms  is  i,  i,  ^,  0,  —^,  —i, 

Examples — 62. 

Find  the  last  term  and  the  sum  of 

1.  2+4  +  6  +  &C.  to  16  terms. 

2.  1  +  3  +  54-&C.  to  20  terms. 

3.  3  +  9  + 15  +  &c.  to  11  terms. 

4.  1  +  8+15  +  &c.  to  100  terms. 

5.  — 5— 3  — 1  — &c.  to  8  terms. 

6.  14-I  +  -I+&C.  to  15  terms. 
Find  the  sum  of 

7.  f +t\  +  tt  +  &c.  to  21  terms. 

8.  4-3-10~&c.  to  10  terms. 

9.  i+f +  1  +  &C.  to  10  terms. 

10.  |_|_i^i_&c.  to  13  terms. 

11.  l  +  2|-  +  4i  +  &c.  to  20  terms. 

12.  |-|4— |i— &c.  to  10  terms. 
276.  By  means  of  the  equations, 

(1)  l=a+{n~iyd,   (2)   S  =  {a  +  l)j, 

and  (3)  S={2a+{n-l)d}^, 

when  any  three  of  the  quantities  a,  d,  I,  n,  S,  are  given,  we 
may  find  the  others. 

We  may  also  use  these  equations  to  solve  many  problems 
in  Arithmetical  Progression. 


ARITHMETICAL   PROGRESSIOI^r.  243 

•  Ex.  1.   The  sum  of  15  terms  of  an  A.  p.  is  600,  and  the 
common  difference  is  5  :  find  the  first  term. 

Since  S  =600,  n=15,  and  d=o,  we  have  by  (3), 

600=(26^  +  14x5)-V-; 
.-.  600=(a  +  35)15;  .*.  a  +  35  =  40;         .-.  a=5. 

Ex.  2.    What  number  of  terms  of  the  series  10,  8,  6,  &c. 
must  be  taken  to  make  30  ? 

^^=30,  ^=10,  d=:-2;  /.  by  (3), 
30  =  [20-2{n-l)]-; 

.-.  {22-27z)^r=:30; 

that  is,  7^'  — 11^=— 30, 

and  the  roots  of  this  quadratic  are  5  and  6,  either  of  which 

satisfies  the  question,  since  the  6th  term  is  0. 

Ex.  3.   How  many  terms  of  the  series  3,  5,  7,  &c.  make 
up  24? 

Here  ;S'=24,  a=3,  ^=2; 

therefore,  by  (3),  24=r[6+2(^-l)]4^; 

whence,,  n  =4,,  or  —6,  of  which  the  first  only  is  admissible 
by  the  conditions  of  the  question. 

277.     Ex.  4.   Find  the  Arithmetical  Mean  between   two 
quantities  a  and  b. 

Let  X  denote  this  mean  ;    then  since  a,  x,  and  b  are  in  A.  P. 

x—a=h—x; 

1  a  +  h 

\v  hence,  x=:———; 

that  is,  the  arithmetical  mean  between  two  quantities  la 
half  the  sum  of  the  quantities. 

Ex.  5.    Insert  five  arithmetical  means  between  11  and  23. 


244  ELEMENTARY    ALGEBRA. 

Here  we  have  to  obtain  an  A.  p.  consisting  of  seven  terms, 
beginning  with  11,  and  ending  with  23. 

Thus,  a=ll,  Z=23,  ^=7; 

therefore,  by  (1),  Art.  276,  23  =  11  +  6^; 

.-.  d=2. 

Thus  the  whole  series  is  11,  13,  15,  17,  19,  21,  23. 

278.     Ex.  6.    The  sum   of  three  numbers  in  a.  p.  is   21, 
and  the  sum  of  their  squares  is  155 :  find  the  numbers. 

Let  X   =    the  middle   number,  and  y  the   common  dif- 
ference; then  x—y,  x,x^-y,  represent  the  three  numbers; 

then  {x—y)-\'X-\-{x^-y)=  21) 

and  {x-^Jy^Vx'-\-(x■\■yy=zlbb ) 

or  reducing, 

^Xz=z    21  ) 

3:?;H2?/^=155  f 
whence,  2;=7,  ^=±2; 

and  the  numbers  are  5,  7,  9. 

Examples — 63. 

1.  The  first  term  in  an  A.  p.  is  2,  the  common  difference 
7,  and  the  last  term  79  :  find  the  number  of  terms. 

2.  The  first  term  of  an  A.  p.  is  13  j-^,  the  common  differ- 
ence — f,  and  the  last  term  |:  find  the  number  of  terms. 

3.  The  first  and  last  of  40  numbers  in  a.  p.  are  1^  and 
If :  find  the  other  terms,  and  the  sum  of  the  series. 

4.  Insert  3  arithmetical  means  between  12  and  20. 

5.  Insert  5  arithmetical  means  between  14  and  16. 

6.  Insert  7  arithmetical  means  between  8  and  —4. 


ARITHMETICAL   PROGRESSIO:^'.  245 

7.  Insert  8  arithmetical  means  between  —1  and  5. 

8.  The  first  term  of  an  arithmetical  progression  is  13, 
the  second  term  is  11,  the  sum  is  40 :  find  the  number  of 
terms. 

9.  The  first  term  of  an  arithmetical  progression  is  5,  and 
the  fifth  term  is  11 :  find  the  sum  of  8  terms. 

10.  The  sum  of  four  terms  in  arithmetical  progression  is 
44,  and  the  last  term  is  17:  find  the  terms. 

11.  The  sum  of  fiye  numbers  in  arithmetical  progression 
is  15,  and  the  sum  of  their  squares  is  55 :  find  the  numbers. 

12.  The  seventh  term  of  an  arithmetical  progression  is  12, 
and  the  twelfth  term  is  7 ;  the  sum  of  the  series  is  171 :  find 
the  number  of  terms. 

13.  A  traveller  has  a  journey  of  140  miles  to  perform. 
He  goes  26  miles  the  first  day,  24  the  second,  22  the  third, 
and  so  on:  in  how  many  days  does  he  perform  the  journey? 

14.  A  sets  out  from  a  place  and  travels  2i  miles  an  hour. 
B  sets  out  3  hours  after  A,  and  travels  in  the  same  direction, 
3  miles  the  first  hour,  3i  miles  the  second,  4  miles  the  third, 
and  so  on :  in  how  many  hours  will  B  overtake  A, 


XXXYII.    Geometrical  Progression. 

279.  Quantities  are  said  to  be  in  Geometrical  Progression 
when  they  proceed  ly  a  comvion  factor  ;  that  is,  when  each 
is  equal  to  the  product  of  the  preceding  by  a  common  factor. 
This  common  factor  is  called  the  common  ratio,  or  simply 
the  ratio. 

Thus  the  following  series  are  in  geometrical  progression : 

Geometrical  Progression. 


246  ELEMEKTARY  ALGEBRA. 

1,  3,  9,  27,  81 ... . 

^>  ^}  h  iQy  it 

__JL    __4      ___l  6 

a,  ar,  ar"^,  &c. 

The  commo7i  ratios  being  3,  ^,  — f,  and  r,  respectively.  The 
common  ratio  is  found  dy  dividing  any  term  hy  the  term 
ivJiicli  immediately  precedes  it;  therefore,  if  the  quantities 
are  alternately  +  and  — ,  the  ratio  is  negative, 

280.  Given  a  tJie  first  term  and  r  the  common  ratio  of  a 
geometrical  series,  to  find  1  the  nth  term,  and  S  the  sum  of 
n  terms. 

Here,  since  a  is  the  first  term  and  r  the  common  ratio,  the 
second  term  is  ar,  the  third  term  is  ar"^,  the  fourth  term  is 
ar^,  and  so  on;  where  the  index  of  r  in  any  term  is  less 
hy  one  than  the  number  of  the  term.  Thus  then  the  ?zth 
term  l—ar^~'^.     (1) 

Again,  8—a^-ar^ar'^-\-kQ,.,     +ar"-^; 

and  TS=.ar-{-ar''-\-ar^-\-kQ,,,     +ar"-Har"; 

therefore,  by  subtraction, 

rS—S^ar'^—a, 
the  other  terms  disappearing.    Hence, 

8— —z=a.^ ^  (2);  ox  B— (3),  smce  rl=iar\ 

Ex.  1.  Find  the  6th  term,  and  the  sum  of  6  terms  of 
1,  2,  4,  &c. 

Here  a—\,   r=2,    n—^\ 

.•.?=1X  2^=32;  and /S'^^^-^  63. 

/C  —  1 

Ex.  2.  Find  the  8th  term,  and  the  sum  of  8  terms  of  81, 

-27,  9,  &c. 

The  common  ratio,  how  found;  when  negative.  To  find  the  /ith  term,  and 
the  sum  of  n  terms. 


GEOMETRICAL    PROGRESSJOIn^  247' 

Here  a— SI,   r=  —  l,   n-. 

therefore^ 


)re,     ?=81x(^|-)'-3*X-^  =  -|-3=:-^; 


and 


;S^=:Ii ^  =  60-2-0- 

—  3— 1 


Ex.  3.  Find  the  sum  of  8  terms  of  the  series,  4,  2,  1,  ^,  (&c. 
•    Here  a—^,    r—^,    n—S; 

therefore,  without  finding  I, 

a    ^V2^"V_^V^~2V_255     2_255 
i-1  1-i  64^1""  32* 

Ex.  4.  Find  the  sum  of  1— f +-^— &c.  to  4  terms. 

Here  a=l,    r=— |,    71  =  4; 


,(-iy- 


4*     .       4^-3* 


-1      o-i-1 


.  ^-IX.^— ^^ -»^I — j^-      3      256-81 

__175___25 

-     7.3' ~     27' 

Ex.  5.  Find  the  sum  of  2^— 1+|— &c.  to  5  terms. 
Here  ^=f?    ^=-"1?    n=6; 


(-1)- 


••^    -^-      _|^i         2-  -1-1      2-      I 

__5^   ^   32  +  3125_3157__ 
""2*  7*        5'       ~14.5^~  ^* 


Examples — 64. 

Find  the  last  term  and  the  sum  of 
1.  1+4  +  16  +  &C.  to4terms.       2.  5 +  20+ 80 +&c.  to  5  terms. 

Formulas  (1),  (2),  (3). 


'ZiS  ELEMEIs^TARY   ALGEBRA. 

3.  3  +  6  +  12  +  &c.to6terms.      4.  2— 4  +  8- &c.  to  8  terms. 
5.  l-4+16-&c.to7terms.       6.  1-2+2'- &c.  to  10  terms. 

Find  the  sum  of 
7.  i  +  i  +  tV  +  &c.  to  8  terms.      8.  i  +  i  +  f  +  &c.  to  6  -terms. 
9.  3 +i_|.2 +&C.  to  6  terms.      10.  3-i  +  -^j—&c.  to  5  terms. 

281.  If  the  terms  of  a  geometrical  progression  decrease 
numerically  as  the  series  proceeds,  then  the  common  ratio  r 
is  a  proper  fraction ;  that  is,  r  is  less  than  1.  Therefore  tho 
powers  r^,  r^  r*,  ....  r'^,  are  still  less  than  1,  and  ar^  less 
than   a.      Both  the   numerator   and    denominator  of   the 

fraction  S= z—  are  then  negative,  and  we  may  write  it 

^    a—af       a        ar^ 


1— r  "1— r    1— r 

ISTow,  the  greater  we  take  the  number  of  terms  n,  the  less 
will  be  the  value  of  ar"" ;  and  therefore,  by  taking  n  suffi- 
ciently great,  we  may  make as  small  as  we  please. 

Hence  by  making  n  sufficiently  great,  we  render  the  value 

of  S  as  near as  we  please. 

1—r  ^ 

This  result  we  enunciate  thus  :  In  a  Geometrical  Progres- 
sion in  loliich  the  common  ratio  is  a  iwojper  fraction,  hy 
talcing  a  sufficient  number  of  terms  the  sum  of  the  series  can 

be  made  to  differ  as  little  as  ive  please  from> . 

is  said  then  to  be  the  Limit  of  the  sum  of  the  series 

1—r 

a,  ar,  ar"^,  &c.,  when  r<l ;  or  we  say,  for  shortness  (but  not 

correctly),  S=z is  the  sum  of  the  series  to  infinity.  Using 

the  same  language,  these  series  are  called  infinite  Geometrical 
Progressions. 

'  When  the  common  ratio  is  a  proper  fraction.  The  limit  of  the  sum  oi  a 
series,  when  r<l.    Infinite  Geometrical  Progressions, 


GEOMETRICAL   PROGEESSIOK".  249 

It  is  common  to  denote  the  Limit  of  such  a  sum  by  2. 
Ex.  1.  Find  the  Limit  of  the  sum  of  the  series  1  +  i  +  i  +  &c. 

Here  a=l,  r=i;  .*.  :s=- — -=—  =  2;   that  is,  the  more 
1— -2-     Y 

terms  we  take  of  tliis  series,  the  more  nearly  will  their  sum 
=  2,  but  will  neyer  actually  reach  it. 

Ex.  2.  Fmd  the  sum  of  2i—^  +  ^—&G.  ad  infinitum. 
Here  a=n,  r^-^,  :.  ^=i3|z^=  1^^=1=2 A- 

282.  Eecurring  Decimals  are  examples  of  infinite  Geo- 
metrical Progressions.     Thus,  for  example, 

.3333  ....  denotes  iV  +  yfo  +  1A-0+&C:, 
a  G.  p.  of  which  the  first  term  <^=yV  ^^^  ^^  ^^2,^0  r=^. 
Hence  we  may  say  that  the  Limit  of  this  decimal  is 

Again;  .3242424  ....  denotes  33_+^|a_  +  .^^_2a__+&c. 
Here  the  terms  after  -^-^  form  a  G.  p.  of  which  the  first 
term  =  y 0^0-0^  ^^^  ^^^  common  ratio  is  y^-g-.     Hence  the 


Limit  of  this  series  is     iQOQ- — ^2^_.    Therefore  the  limiting 

^        Too" 

value  of  the  recurring  decimal  is 

3_       24  _  3x99  +  24      3  (100-1) +  24 _  324-3^ 
10"^  990  ~        990        ""  990  ~     990    ' 

and  this  value  accords  with  the  rule  in  Arithmetic.     (See 

Venable's  Arithmetic,  Eecurring  Decimals.) 


Examples — 65. 
Find  the  Limits  of  the  sums  of  the  following  series : 
L  4+2  +  1+&C.        2.  |  +  ^  +  |  +  &c.      3.  i— iV+A  +  <^^'- 

Recurring  Decimals. 

11* 


250  eleme:s'taky  algebra. 

4.  I-H-I-&C.       5.  l-i  +  i-&c.      6.  l-|  +  ^-&c. 

Find  the  limiting  values  of  the  following  recurring 
decimals : 

10.  .151515 11.  .123123123 12.  .4282828 

283.  By  means  of  the  equations  of  Geometrical  Progres- 

,        „  1       rv    Ir—a     af  —  a               a 
Bion,  yiz.,     l—ar''-^.     8— -= -,     :2=- ,  we  may 

solve  many  problems  respecting  series  of  this  kind.  It  is  not, 
how^ever,  generally  easy  to  find  7i  from  the  other  quantities, 
because  it  is  an  exponent.  The  method  of  logarithms  will 
serve  to  find  it  in  all  cases. 

Ex.  1.  Find  a  geometrical  series,  whose  1st  term  is  2  and 
7th  term  -^j. 

Here  a=2,  1=-^,  n=7;  ,-, -^=2r%  and  r°=-g^,  whence 
r=dt:^,  and  the  series  is  2,  ±1,  i,  ±J,  &c. 

Ex.  2.  Given  6  the  second  term  of  a  geometrical  series  and 
54  the  fourth,  to  find  the  first  term. 

54    dT^ 
Here      6=ar,      54=ar^:       -'- -yr— — ,  or  9=r^: 

6      ar 

n 

hence  r=:dt3,       a=  —  =  ±:2, 

r 

Ex.  3.     Insert  three  geometrical  means  between  2  and  32. 

Here  we  have  to  obtain  a  geometrical  progression  con- 
sisting of  five  terms,  beginning  wath  2  and  ending  with  32. 
Thus,  a=2,  1  =  32,  n=5;  therefore, 

32=2r^;     .-.  r'=16;     .-.  r=2. 

Thus  the  series  is  2,  4,  8,  16,  32. 

Ex.  4.  How  many  terms  of  the  series  2,  —6,  18,  &c.,  must 
be  taken  to  make  —40  ? 


GEOMETRICAL   PROGRESSIO:^'.  251 

Here  a=:2;     r=-3;     S=-4:0; 


therefore, 


-3-1     ' 


hence  2(— 3)"=162; 

/.  (_3)^==:81. 
But  we  know  that  81=3*;  therefore,  n=4:. 

Examples — 66,  • 

1.  Insert  3  geometrical  means  between  1  and  256. 

2.  Insert  4  geometrical  means  between  5^  and  40-|-. 

3.  Insert  4  geometrical  means  between  3  and  —729. 

4.  The  sum  of  three  terms  in  geometrical  progression  is 
63,  and  the  difference  of  the  first  and  the  third  term  is  45 : 
find  the  terms. 

5.  The  sum  of  the  first  four  terms  of  a  geometrical  pro- 
gression is  40,  and  the  sum  of  the  first  eight  terms  is  3280 : 
find  the  progression. 

6.  The  population  of  a  country  increases  annually  in  G.  p., 
and  in  four  years  was  raised  from  10,000  to  14,641  souls :  by 
what  part  of  itself  was  it  annually  increased  ? 

7.  The  sum  of  an  infinite  geometrical  series  is  3,  and  the 
sum  of  its  first  two  terms  is  2f :  find  the  series. 

8.  The  sum  of  an  infinite  geometrical  series  is  2,  and  the 
second  term  is  —  | :  find  the  series. 

9.  A  body  moves  through  20  miles  in  the  first  one-mil- 
lionth part  of  a  second,  18  miles  in  the  second  millionth 
part  of  a  second,  and  16|  miles  in  the  third  millionth  part 
of  a  second,  and  so  on  forever:  what  is  the  limit  of  the  dis- 
tance from  its  point  of  departure,  which  it  can  attain  ? 


252  ELEMENTAEY   ALGEBRA. 

XXXVIII.     Harmonical  Progression. 

284.  Quantities  are  said  to  be  in  Har7nonical  Progression 
when  their  reciprocals  are  in  A.  p.  Thus,  since  1,  3,  5,  &c., 
^,  —i,  —  f,  &c.,  are  in  aiithmetical  progression,  their  recip- 
rocals, 1,  ^,  ^,  &c.,  2,  —2,  —  f,  &c.,  are  in  Harmonical  Pro- 
gression. 

The  term  Harmonical  is  apj)lied  to  series  of  this  character 
from  the  fact  that  musical  strings  of  equal  thickness  and 
tension  will  produce  harmony  when  sounded  together,  if 
their  lengths  be  as  the  reciprocals  of  the  arithmetical  series 
of  the  natural  numbers  1,  2,  3,  4,  &c. 

285.  If  three  quantities,  A,  B,  and  C,  are  i7i  Harmonical 
Progression,  tlien  2uiU  A  :  0 : :  A—B  :  B—  C, 

For,  by  definition,  -j,  ^ ,  and  -^  are  in  arithmetical  pro 
gression;  therefore, 

1   2_A  i. 

B^A^G'S' 
multiplying  by  ^^(7,        AG-BG=AB-AO; 
that  is,  G {A-B)=A  (B-G) ; 

therefore.  A:  G::A—B:B—G,  wliicli  loas  to  le  ^proved. 

This  property  is  sometimes  given  as  the  definition  of  Har- 
monical Progression,  and  the  property  gi^^en  as  the  defini- 
tion in  Art.  284  deduced  from  it. 

286.  We  cannot  find  a  convenient  expression  for  the  sum 
of  any  number  of  terms  of  a  harmonical  series ;  but  many 
problems  with  regard  to  such  series  may  be  solved  by  invert- 
ing the  terms,  and  then  treating  these  reciprocals  as  in  arith- 
metical progression. 

Harmonical   Progression.     Demonstrate   Art.  285.     To    fin(^    the   sum   of  the 
terms  of  a  Harmonical  Progression. 


HAEMOKICAL   PROGRESSION.  253 

Ex.  1.  Continue  to  three  terms  each  way  the  H.  p.,  2,  3,  6. 

Here,  since  -J-,  i,  ^  are  in  a.  p.  with  common  difference 
— I",  the  arithmetical  series  continued  each  way  is 

1.  h  h  h  h  h  ^-h  -i 
therefore  the  harmonical  series  is 

1,1,1,2,3,  6,  infinity,  -6  -3. 
Ex.  2.  Insert  five  harmonical  means  between  f  and  -fj. 

Here  we  have  to  insert  five  arithmetical  means  between  | 
and  -1^^-.  Hence  by  equation  (1)  Art.  275,  -i/ =1  +  6(7;  there- 
fore, 6^— f,  and  d=-^-^''y  hence  the  A.  p.  is 

t     2.A     2_6     2  7     J.8.     2JL    JL5_  • 
^i    165    16,    iti    165    165     8    ? 

and  therefore  the  H.  p.  is 

287.  The  geometrical  mean  G  between  two  quantities  a 
and  l  is  the  geometrical  mean  between  their  arithmetical 
mean  A  and  their  harmonical  mean  H, 

Eor,  the  arithmetical  mean  between  a  and  l  is  (Art.  277), 

A="^.     (1) 

h       P 
The  geometrical  mean  6^  gives --^=—;  .*.  G'^a'h  and  G—^/ab, 

To  find  the  harmonical  mean  ZTwe  have  ———^=-- ; 

h     H     H     a 

'%ah 
therefore,  aE—al—al  —  lH.     or  H— — -7:     (2) 

a-\-o 

multiplying  (1)  and  (2),  we  get 

AH———X ^  —  al—G\ 

Therefore  G—s/ AH,  or  G  is  the  geometrical  mean  be- 
tween A  and  H, 

Demonstrate  Art.  287. 


254  ELEMEKTAKY   ALGEBRA. 


Examples— 67. 

1.  Continue  the  Harmonical  Progression  6,  3,  2  for  three 
terms. 

2.  Continue  the  Harmonical  Progression  8,  2,  l^-  for  three 
terms. 

3.  Insert  2  harmonical  means  between  4  and  2. 

4.  Insert  3  harmonical  means  between  —  and  — . 

o  /vL 

5.  The  arithmetical  mean  of  two  numbers  is  9,  and  the 
harmonical  mean  is  8 :  find  the  numbers. 

6.  The  geometrical  mean  of  two  numbers  is  48,  and  the 
harmonical  mean  is  46^2-g-:  find  the  numbers. 

7.  Find  two  numbers,  such  that  the  sum  of  their  arith- 
metical, geometrical,  and  harmonical  means  /j  9|,  and  the 
product  of  these  means  is  27. 

8.  Find  two  numbers,  such  that  the  product  of  their  arith- 
metical and  harmonical  means  is  27,  and  the  excess  of  the 
arithmetical  mean  above  the  harmonical  mean  is  1|-. 


XXXIX.    Pekmutations  and  Combinations. 

288.  The  Permutatio7is  of  any  number  of  things  are  the 
different  arrangements  which  can  be  made  of  them  by 
placing  them  in  different  orders,  taking  either  all  the  things, 
or  a  certain  number  of  them  at  a  time,  together. 

Thus  the  'permutations  of  a,  Z*,  c,  taken  all  together,  are 
ale,  achy  hca,  cia,  cah,  lac ;  taken  two  together,  are  ac,  ca,  al^ 
la,  Ic,  cl. 

289.  Note. — Some  writers  on  Algebra  restrict  the  word  permufa- 
tions  to  the  case  where  the  things  are  taken  all  together,  and  call  tlie 


PERMUTATIOIsrS   A^^D    COMBII^TATIONS.  255 

sets  in  all  otiier  cases  Variations^  or  Arrangements.  But  this  distinction 
is  not  always  observed,  and  we  shall  use  the  word  permutations  in  all 
cases. 

290.  The  number  of  permutations  of  n  tilings,  tciken  two 
together,  is  n  (n— 1) ;  taken  three  together,  ^5  n  (u  — 1)  (n— 2). 

Let  there  be  n  different  things,  a,  i,  c,  d,  &c.  Remove  one 
of  them,  a-,  there  will  be  ^—1  things,  h,  c,  d,  &c.,  left;  now 
place  a  before  each  of  these  n—1  things;  we  thus  get  n—1 
permutations,  n  things  taken  tiuo  together,  in  which  a  stands 
first.  JSText  remove  d  from  the  n  things ;  there  will  remain 
oi—l  things,  a,  c,  d,  &c. ;  and  placing  b  before  each  of  these 
we  get  71—1  permutations  of  n  things  tahen  two  together,  in 
which  b  stands  first.  Similarly  placing  c  before  each  one  of 
the  other  letters,  we  find  n—1  permutations,  in  w^hich  c 
stands  first;  and  so  on  for  the  rest.  Therefore,  on  the 
whole,  there  are  n{7i—l)  permutations  of  n  things  talcen  two 
together,  or  ttuo  and  ttoo,  as  is  the  usual  phrase.  Therefore 
there  are  also  {n—1)  {^^— 2)  permutations  of  n—1  things 
taken  tioo  together. 

Let  now  a,  one  of  the  n  things,  be  again  removed ;  the  re- 
•maining  n—1  things,  by  what  w^e  have  just  proved,  gives 
{n  —  1)  {n—2)  permutations  when  taken  tivo  together;  put  a 
before  each  of  these  permutations ;  we  thus  get  {7i  —  l){7i—2) 
permutations,  each  composed  of  three  things,  in  w^hich  a 
stands  first.  Similarly,  there  are  {7i—l)  (^^— 2)  permuta- 
tions each,  of  three  things  in  which  b  stands  first ;  similarly, 
there  are  as  many  in  which  c  stands  first,  and  so  on  for  the 
rest.  Therefore  there  are  72  {n  —  1)  {71— 2)  per77iu  tat  ions  of  7i 
thiiigs  talcen  three  together, 

291.  We  observe  at  once  that  the  second  term  of  the  -usi} 
factor  of  the  product  which  expresses  the  number  of  permu- 
tations in  each  case  of  the  preceding  article,  is  numerically 
less  by  one  than  the  7iumber  of  things  taken  together.      From 

Permutations.    Demonstrate  Art.  290. 


256  ELEMENTARY   ALGEBRA. 

these  cases  we  miglit  infer  by  induction  that  this  is  a  general 
law,  and  that  the  number  of  permutations  of  7i  letters  taken 
r  together  is  n{;n—l)  {n—2)  ....  (?^— r+l),  and  this  we 
can  now  demonstrate. 

For,  suppose  this  law  to  hold  for  the  number  of  permuta- 
tions of  n  things,  a,  l,  c,  d,  &c.,  taken  r—1  together,  which 
would  therefore  be 

71(71-1)  (n-'^)  ....  (^^-(r-l)+l). 

]^ow  leaye  out  a\  there  will  be  7^— 1  things,  h,  c,  d,  &c., 
and  the  permutations  of  these,  taken  r—1  together,  will  be, 
by  the  preceding  result, 

{71-1)  {n-2) {7i-l-[r-l)+l); 

that  is,     {71-1)  {71-2) {7i-r+l). 

Set  a  before  each  of  these  permutations;  we  get  thus 
(^—1)  (n— 2) {7i—r-\-l)  permutations  taken  r  to- 
gether, in  wiiich  a  stands  first.  Similarly,  we  have  as  many 
in  which  b  stands  first ;  as  many  in  wiiich  c  stands  first,  and 
so  of  the  rest ;  therefore  on  the  whole  there  would  be, 

n{7i-l)  {71-2) {7i-r+l) 

permutations  of  7i  things  taken  r  together.^ 

If  then  the  formula  holds  when  the  7i  things  are  taken 
r—1  together,  it  will  hold  when  they  are  taken  r  together; 
but  it  has  been  proved  true  when  they  are  taken  3  together ; 
it  holds,  therefore,  when  they  are  taken  4  together ;  and 
therefore,  when  taken  5  together,  and  so  on ; 

that  is,  7i{7i  —  l)  {71—2) {7i—r+l) 

represents  the  Ttwmher  of  per7nutations  of  ti  tilings  taken  r 
togetlie7%  for  all  values  of  r  (these  values  being  limited  only 
by  the  definition). 

General  law  for  the  number  of   Permutations  of  n  letters  taken  r  together. 


PERMUTATIONS   AND   COMBUST ATIONS.  257 

292.  Hence,  denoting  by  P^,  P^,  P^,  &c.,  P^,  the  number 
of  permutations  of  n  things  taken  1,  2,  3,  &c.  r,  together, 
we  have  from  the  preceding  formula, 

P^^n,  P^==n(n-1),  P,=n{n-l){n-2),  &c. 

P,,=n{n—1) (^— r  +  1). 

29B.  If  T=7i,  that  is,  if  all  the  quantities  are  taken  to- 
gether, then  the  number  of  permutations  (P)  of  n  things  is 

n{n—l)  (71—2) (n—n  +  l); 

that  is,  n{n—l)  {n—2)  .' .  .  1; 

or  reversing  the  order  of  the  factors,  we  have, 

P=1X2X3  .  .  .  .  Xn. 

This  result  we  may  enunciate  thus : 

The  number  of  Permutations  of  n  things,  taken  all  together, 
is  equal  to  the  product  of  the  natural  numbers  from  1  to  n, 
inchisive. 

Thus  the  number  of  permutations  of  8  letters,  taken  all 
together,  is  1x2x3x4x5x6x7x8. 

294.  For  the  sake  of  shortness,  the  continued  product, 
1.2.3.4  .  .  .  .  n,  is  often  denoted  by  [^;  thus  [^  denotes 
the  product  of  the  natural  numbers,  from  1  to  ^  inclusive. 
The  symbol  \n  may  be  read  factorial  n. 

Ex.     [8,  (read  factorial  eighty,  denotes  the  product 

1X2X3X4X5X6X7X8. 

295.  To  find  the  number  of  permutations  of  n  things, 
which  are  not  all  different,  taken  n,  i,  e,  all  together. 

Express  the  formula  by  the  use  of  the  symhols  Pi,  P21  Pz^  ^tc.    The   nuii> 
ber  of  Permutations  where  r=:-n.    The  symbol  \n.      Demonstrate  Art.  295 


258  ELEMEis^TARY    ALGEBRA. 

Let  there  be  n  letters ;  and  suppose  jp  of  them  to  be  a's,  q 
of  them  to  be  ^'s,  r  of  them  to  be  c's,  &c. ;  the  number  of  per- 
mutations of  them,  taken  all  together,  will  be, 

1.2.3  .  .  .  .  ^ 


1.2.3  ....  ^Xl.2.3  ....  ^X&c*     ,     . 

For  let  N  be  the  number  of  such  Permutations.  Suppose 
now  that  in  any  one  of  them  we  change  the  ^  a's  into  dif- 
ferent letters;  then  these  letters  might  be  arranged  in 
1.2.3....^  different  ways,  and  so  instead  of  this  one 
permutation,  in  which  p  letters  would  have  been  «'s,  we 
shall  now  haye  1.2.3  .  .  .  .  p  different  permutations.  The 
same  would  be  true  for  each  of  the  iV^  permutations ;  hence, 
if  the  p  a's  were  all  changed  to  different  letters,  we  should 
have  all  together  1 . 2 .  3  ....  j^X indifferent  permutations 
of  n  letters,  whereof  still  q  are  5's,  r  are  c's,  &c. 

So,  if  in  these  the  q  b's  were  changed  to  different  letters, 
we  should  have  1.2.3  ....  gXl.2.3  .  .  .  .^XiV" differ- 
ent permutations  of  n  things,  whereof  still  r  would  be  c's ; 
and  so  we  may  go  on,  until  all  the  n  letters  are  different. 
But  when  this  is  the  case  we  know  that  their  whole  number 
of  permutations  ==1.2.3  .  .  .  .  n.     Hence, 

1.2.3  ...  .i?Xl.  2.3  ...  .  gX&c.XiV'=1.2.3  .  .  .  n, 

1.2.3  .  ,  .  .  n 

and       -^-12,3 i?Xl.2.3  ....  gX&e.' 

This  value  of  iV^  may  be  written  by  the  notation  of  Art. 
294;  thus, 

\n 

Ex.  1.  How  many  changes  can  be  rung  with  5  bells  out 
of  8  ?     How  many  with  the  whole  peal  ? 

Here  P,  =38.7.6.5.4=6720;  P=8.7.6.5.4.3.  2  1  =  40320. 


PERMUTATIOJ^^S   Al^D    COMBIJsTATIOKS.  259 

Ex.  2.    How  many  differ tnt  words  may  be  made  with  all 

the  letters  of  the  expression  a^Jfc  ? 

12  3  4  5  6 
Here  are  6  letters,  3  a's,  and  2  Z>'s ;  .-.  N=   '  '  '  '  '  =  60. 

JL./V.d  X  1"V 


Examples — 68. 

1.  How  many  changes  may  be  rung  with  5  bells  out  of 
6,  and  how  many  with  the  whole  peal  ? 

2.  In  how  many  different  orders  may  7  persons  seat 
themselves  at  table  ? 

3.  How  many  different  words  may  be  made  of  all  the  let- 
ters of  the  word  laccalaureus  ? 

4.  How  many  different  words  may  be  made  of  all  the 
letters  of  the  word  Mississippi? 

5.  How  many  different  words  may  be  made  of  all  the 
letters  of  the  word  Alabama? 

6.  Of  what  number  of  things  are  the  permutations  720  ? 

7.  There  are  7  letters,  of  which  a  certain  number  are  a^% ; 
and  210  different  words  can  be  made  of  them :  how  many  a's 
are  there  ? 

8.  If  the  number  of  permutations  of  n  things  taken  4 
together  is  equal  to  twelve  times  the  number  of  permu- 
tations of  n  things  taken  2  together,  find  n, 

296.  The  Comlinations  of  any  things  are  the  different  col- 
lections or  sets  that  can  be  made  of  them,  without  regarding 
the  order  in  which  the  things  are  placed.  Thus  the  com- 
binations of  a,  b,  c,  taken  two  together,  are  ab,  ac,  bc\  of  a, 
b,  c,  d,  three  together,  are  abc,  abd,  acd,  bed. 

297.  It  is  readily  seen  that  each  combination  which  con- 
tains  r  things   will  furnish   1.2.3 r  permutations  of 

Combinations.    Demonstrate  Formula  (Art.  298). 


260  ELEMENTAKY    ALGEBIcA. 

• 

these  things  taken  all  together.  For  we  haye  seen,  Art.  291, 
that^z  things  give  1.2.3  , . ,  ,  n  permutations. 

Thus  the  combination  abc  supplies  1.2.3,  or  6  permuta- 
tions, abc,  act,  lac,  lea,  cal,  da, 

298.  The  numler  of  comlinations  of  n  different  tilings, 
talcen  r  together,  is 

n  (^—1)  (^—2)  ....  (?^— r+1) 
1.2.S r  * 

For,  each  combination  of  r  things  will  supply  1.2.3  . .  . .  r 
permutations  of  r  things ;  hence,  if  Cr  denotes  the  number 
of  combinations  of  n  things,  r  together,  we  have 
1.2.3  ....  rxC^=number  of  permutations  of  n  things,  r 
together, 

=  y^—n  {n—\)  (^--2) (n—r  +  l) ; 

^  _n{n—l)  {n—2)  ....  (n—r  +  l) 
.-.  G,-  1.2.3  .  . . .  r  • 

Therefore,   (7i=— ,  C^=^  ^^' ,  0,=^ — ^23 '  ®^-' 

where  Ci,  Co,  C^,  &c.,  express  the  number  of  combinations  of 
n  letters  taken  one  and  one,  two  together,  three  together,  &c. 

«Aa    rn\.  '        n      n{n-l){n-2) (^-r+1) 

299.  The  expression    Cr  —  -^ \  9,^^ ~^^ 

may  be  put  in  a  yery  conyenient  form ;  for,  by  multiplying 
the  numerator  and  denominator  of  the  aboye  fraction 
by  1.2.3  ....  {n—r),  it  becomes 

n  jn-l)  {n-2) (n-r  +  l)  X  (n-r) 3.2.1 

1.2.3 r  X  1.2.3....  (^-r) 

1.2.3 n  lH: 


"1.2.3 r  X  1.2.3 (?z-r)~"  [r  | 


PERMUTATIONS   AND    COMBINATlOjSfS.  201 

■'■  <^ -7-^-        (1) 
\r  \7i  —  r 

300.  The  number  of  comUnations  of  n  things  taken  r  to- 
gether is  the. same  as  the  ^lumber  of  combinations  ofn  things 
taken  n— r  together. 

For,  to  find  the  number  of  combinations  of  n  things  taken 

n—r  together,. we  simply  write  n—r  for  r  in  formula  (1). 

We  get  thus 

\n  \n 

'^''~  \n—r\7i—{n—r)^\n—r\)r_^ 

which  is  equal  to  Cr ,  which  was  to  be  proved. 

The  truth  of  this  proposition  is  also  evident  from  a  very 
simple  consideration,  viz.,  that  when  we  take  r  things  from 
^^  things,  n—r  things  will  be  left;  and  for  every  different 
collection  containing  r  things  there  will  be  a  different  col- 
lection left  containing  n—r  things;  therefore  the  number  of 
the  former  collections  must  be  equal  to  that  of  the  latter. 

Ex.  1.  Eequired  the  number  of  combinations  of  20  things 
taken  18  together. 

Here  the  number  of  combinations  of  20  things  taken  18 
together  is  equal  to  the  number  of  combinations  of  20  things 
taken  2  together, 

that  is,  (7is=C,-=^f^=:10xl9  =  190. 

Ex.  2.  Eind  the  number  of  combinations  of  10  things,  3 

and  6  together. 

^  ^     10.9.8     ^__        Ann     10.9.8.7     ^,^ 

Here       C,=^-^j^=120,  and  C,=  C=  ^  2.3.4  ^ 

Ex.  3.  How  many  words  of  6  letters  might  be  made  out  of 
the  first  10  letters  of  the  alphabet,  with  two  vowels  in  each 
word  ? 

state  th**  r^Tinciple  explained  in  Art.  300. 


262  ELEMEKTARY    ALGEBRA. 

In  these  10  letters  tliere  are  7  consonants  and  3  vowels; 
and  in  each,  of  the  required  words  there  are  to  be  4  conso- 
nants and  2  vowels :  now  the  7  consonants  can  he  combined 
four  together  in  35  ways,  and  the  3  vowels,  two  together,  in 
3  ways;  hence  there  can  be  formed  35x3  =  105  different  sets 
of  6  letters,  of  which  4  are  consonants  and  2  vowels :  but 
each  of  these  sets  of  6  letters  may  be  permuted  6.5.4.3.2.1 
=  720  ways,  each  of  these  forming  a  different  word,  though 
the  whole  720  are  composed  of  the  same  6  letters ;  hence  the 
number  required=105X  720=75600. 


Examples — 69. 

1.  How  many  combinations  can  be  made  of  9  things,  4 
together  ?  how  many,  6  together  ?  how  many,  7  together  ? 

2.  How  many  combinations  can  be  made  of  11  things,  4 
together  ?  how  many,  7  together  ?  how  many,  10  together  ? 

3.  A  person  having  15  friends,  on  how  many  days  might 
he  invite  a  different  p'arty  of  10  ?  or  of  12  ? 

4.  Fintl  the  number  of  combinations  of  100  things,  taken 
98  together. 

5.  Four  persons  are  chosen  by  lot  out  of  10 :  in  how  many 
ways  can  this  be  done?  on  how  many  of  these  occasions 
would  any  given  man  be  taken  ? 

6.  The  number  of  combinations  of  ^+1  things,  4  to- 
gether, is  9  times  the  number  of  combinations  of  n  things,  2 
together :  find  n. 

7.  How  often  may  a  different  guard  be  posted,  of  6  men 
out  of  60  ?  on  how  many  of  these  occasions  would  any  given 
man  be  taken  ? 

8.  How  many  words  may  be  formed,  each  consisting  of 
three  consonants  and  a  vowel,  out  of  19  consonants  and  5 
vowels. 


Bi:ts"OMIAL   TIIEOKE^.!.  ^  263 

XL.      Binomial  Theorem. 

301.  The  Bi7iomial  Theorem  is  the  name  given  to  a  rule 
discovered  by  Sir  Isaac  JSTewton,  by  means  of  which  any  bi- 
nomial may  be  raised  to  any  given  power  much  more  expe- 
ditiously than  by  the  process  of  repeated  multiplication  given 
in  Involution. 

302.  To  prove  the  Binomial  Theorem  when  the  index  of  the 
poiver  is  a  positive  ivhole  ntimher,     (Bobillier's  Proof.) 

By  actual  multiplication  the  successive  powers  of  the  bi- 
nomial a-\-x  are  found  to  be  as  follows : 

{a-\-xy=a-{-xi 

{a-\-xy=:a^-\-2ax-{-x^ ; 

{a+xy=a^  +  da'x-\-dax'+x'' ; 

(a  +  a;)  * = a'  +  4a'a; + 6aV  +  4:ax^  +  x* ; 

which,  by  dividing  the  first  by  1,  the  second  by  1.2,  the  third 
by  1.2.3,  the  fourth  by  1.2.3.4,  and  using  the  factorial  nota- 
tion of  the  preceding  chapter  to  denote  the  continued  pro- 
ducts 1.1,  1.2,  1.2.3,  &c.,  may  be  written  thus : 

(a  +  xy  _a'      x^ 

[1— |i  +  li' 

(a  +  xy  _  a^      g'  eg'      x^ 

{a^xy      a^       a^  x"      a'  x^      x^ 

[3      ""]3  ^  l^'^"[ili^]3  ' 

{a  +  xy      a'      a^  x'     ^^      a'  x""      o^ 

in  which  a  laio  of  formation  is  easily  perceived  in  relation 
to  the  exponent  of  the  power  of  the  binomial.     The  same 

Binomial  Theorem.    Proof  of  the  Binomial  Theorem,   when  the  index  of  the 
power  is  a  positive  whole  number. 


264  ELEMENTARY    ALGEBRA. 

law  of  formation  of  the  terms  of  the  expansion  is  found  to 
hold  for  {a  +  xy,  (a-\-xy,  &c.  Now  the  introduction  of  the 
new  factor  «  +  a;  in  order  to  convert  {a  +  xY~^  into  (a  +  ic)% 
inyolves  precisely  the  same  processes  as  the  introduction  of 
the  same  factor  {a  +  x)  to  conyert  {a  +  xY  into  {a-\-xy.  It 
is  reasonable  then  to  assume  that  if  the  law  is  true  for 
{a-\-xy'~^,  it  is  true  for  [a  +  xY]  now  we  know  by  actual 
multiplication  it  is  true  for  {a-\-xy\  hence  it  is  true  for 
{a-{-x)\  and  hence  for  {a-{-xy,  &c.  Therefore  the  law  holds 
generally — viz.,  for  any  positive  whole  number  exponent 
we  have 

{x-\-aY  _  cc**       a  x""-'         a^  x"^""        a^  x""-^  a"  ^ 

\n      ~~\n'^\l  |/^-"l"^]2~j^^^"^  |3  \n-3  "^"j^ ' 

which  may  be  written, 

\n  \n 

I.  ix  +  aY=--x''  +  TT-T^— ;^^''~'  +- 


[1   \n-l  [2  1^-2 

\n  \n 

+  ,^  ,       ^  6^V-^  + +  — J==—  a'x^^. . .  ,'\-a\ 

[3  1^—3  [r    \n  —  r 

Or  by  cancelling  the  like  factors  in  the  coefficients, 

II.  {x  +  aY=x''-{-  nax""-' + — ^ — ^  a^x'"-'  +  —^ —^ ^ 

[±  li 

[r 
303.  In  these  expressions  I.  and  II.  the  corresponding  terms 


\n 


Ir    In— 


aV-'     (1) 


and  nin-l){n-2)  (n-r+l)^^^^^ 

are  the  same,  and  they  express  the  term  which  has  r  terms 
before  it;  that  is,  the  (r  +  1)*^  term.     This  term  is  called  the 


BINOMIAL  THEOREM.  265 

General  Term;  and  both  forms  of  it,  (1)  and  (2),  should  be 
carefully  noted  and  remembered. 

304.  If  the  binomial  is  written  {a-\-xy,  the  expansion  I. 
would  be 

\n  \n 

li  r^~^  1^  |y^— 2  ^ 

\n 
the  general  term  being    ,  ^    ~_—  a^-V,  the  exponents  of  a 

and  X  being  interchanged.     Similarly,  II.  becomes 
{a-{-xy=a''  +  na'^''x-\'^\~   ^  a^-V+ 

+  — ^^ — -7 ^ — -^  a^-^a;^+ x\ 

And  if  «^=1,  this  last  giyes 

IIL(l  +  ^)-l+n^+^A'+^^^-^)^^-!)^'.... 
If  l£. 

w  (w-1)  (w-2) ....  (n-r+l)  ^,    ^„ 

305.  I.  The  law  of  the  exponents  of  the  terms  in  the  ex- 
pansion of  the  binomial  formula  is,  that  the  exponent  of  the 
leading  letter  of  the  binomial  is  in  the  first  term  n,  and  that 
of  the  second  letter  is  o ;  and  the  former  decreases  by  unity 
and  the  latter  increases  by  unity,  in  each  successiye  term  to 
the  last  term,  in  which  the  exponent  of  the  leading  letter  is 
0,  and  that  of  the  second  letter  is  n;  and  the  sum  of  the  ex- 
pone7its  of  the  tiuo  letters  in  any  term  is  always  n,  the  ex- 
ponent  of  the  Mnomial.  Moreover,  the  exponent  of  the 
second  letter  in  any  term  expresses  the  number  of  terms 
which  precede  that  term,  and  the  exponent  of  the  leading 
letter  expresses  the  number  of  terms  which  follow  it.     Thus 

The  law  of  the  exponents  of  the  terms. 

12 


260  ELEMEKTARY    ALGEBRA. 

we  can  easily  write  the  exponents  of  the  letters  in  any 
required  term. 

II.  The  numerical  coefficients  of  the  first  and  last  terms  are 
1 ;  the  coefficient  of  the  second  term  is  7i,  or  the  number  of 
combinations  of  7i  things  taken  singly ;  the  coefiicient  of  the 
third  term  is  the  number  of  combinations  of  7i  things  taken 
2  and  2,  &c.,  &c.;  the  coeflS.cient  of  the  (r  +  1)*^  term  is  the 
number  of  combinations  of  n  things  taken  r  together.  And 
since  the  coefficient  of  the  term  which  has  r  terms  before  it 

is  -] — , ,  and  the  coeflBcient  of  the  term  which  has  r 

vr   \n—T 

\n 

terms  after  it  or  n—r  terms  before  it  is  -i — =^— ,  it  follows 

m—r  [r 

that  the  numerical  coefficients  of  any  tivo  terms  equidistant 
from  the  heginning  and  end  are  the  same, 

306.  From  the  above  it  will  be  seen  that  to  find  the  coeffi- 
cient of  any  term  we  may  use  either  of  the  following  rules : 

KuLE  I. — The  coefficient  of  any  term  is  the  exponent  of  the 
hijiomial,  taken  factorially,  divided  hy  the  product  of  the  ex- 
ponents of  the  ttvo  letters  in  that  term,  taken  also  factori- 
ally,  i.  e.,  divided  hy  the  product  of  the  number  of  terms  tvhich 
precede  it  and  the  number  of  terms  lohich  follow  it,  taken 
factorially. 

Examining  Expansion  II.,  Art.  302,  we  have  for  finding 
the  coefficient  of  any  term  from  the  preceding  term, 

KuLE  II. — Multiply  the  coefficient  of  the  preceding  term  hy 
the  exponent  of  the  leading  letter  in  that  term,  and  divide  the 
product  hy  the  numher  of  terms  ivhich  precede  the  required 
term, 

NoTE.-^The  first  rule  is  used  always  when  we  wish  to  find  any  terra 
without  finding  the  preceding  terms. 

Since  from  Art.  305  all  the  coefficients  after  the  middle 
term,  or  (first  middle  term   when  there   are   two),   repeat 

To  lind  the  coeflScient  of  any  term,— Rule  I.  and  Rale  11. 


BIK03<IIAL  THEOREM.  267 

tliemselves,  after  having  found  all  tlie  terms  as  far  as  the 
middle,  we  may  for  the  remaining  terms  simply  write  down 
the  coefficients  already  found,  in  an  inverted  order,  as  in  the 
following  examples. 

Ex.  1.    {a  +  xy 

=:««+  Jl_  a^a;+_[|_  aV+Ji_  aV  + Jl__  aV  +  «&c.: 

1^  m  l\i  liii 

or, 
/    ,    N8      8,8,      8.7    ,,,  8.7.6   .  ,     8.7.6.5  ,  .     , 

= a' + Sa'x  +  28aV + 5  6aV + 70a*x' + 5  6aV 

+28a'x'  +  Sax''+x\ 

Ex.  2.    {a+xy  =  a''+^  a'x+1^1  aV  +  ^l^^a*x'+&c., 

=a'-{'  la'x+^laV  +  d6a'x'+36a'x'+21aV  +  7ax'  +  x\ 

307.  If  the  second  term  of  the  binomial  is  negative,  the 
second  term  of  the  expansion  is  negative,  and  every  alternate 
term  also  negative,  as  is  evident  by  the  rule  of  signs  of 
powers.     Thus, 

{a-xy  =  a'-la'x^2la'x^-3^aV-\-3ba'x'-2la^x'-\-lax'-x\ 

7.6    ,     7.6^ 

=  l--'ix-\-Ux'-3^x'-hdbx'-2lx'  +  W-x\ 
Ex.4.    {3x-\yy 

6  ..  ..,,   ^  .  6.5,.  ,_,   ,,     6.5.4, 


Ex.3.    (l-«jy  =  l-4-:c  +  f^rc^--|:^a;^  +  &c. 


=(3^)^- Y  (3^)^(1^)+^  (3a:)*  {^yy-^^xy  a^)'+&c. 

=  729a;''-6x243a;^Xi^+15x81:z:*Xiy'-20x27a;'Xi^y' 

+  15x92:^XTV^*-6x3a;X^^^*  +  A/ 
=  729a;'  -  729a;>  4-  ^-^^x'f  -  ^\^x^f  +  W^>* "  -h^f  +  A^'- 
Ex.  5.    Find  the  8th  term  in  the  expansion  (x.-{-ay^ 


268  ELEMEIS-TARY    ALGEBRA. 

The  exponents  of  8tli  term  give  x^a\ 

|11 
Hence  the  term  is         -r^ — rr-  x'a^ ; 

or,  cancelling  out  like  factors, 

Ex.  6.   Find  the  middle  term  of  {a—by\ 
The  middle  term  is  the  7th.    Hence  it  is 

112 


|6    |6 


Examples — 70. 

1.  {l  +  x)\        2.  (a-dxy.        3.  {1-xy.        4.   (a-xy, 
5.  (1+^)'^       6.  (1-2^)^^         7.  {a-3xy.      8.  (2x+ay. 
9.  (2«-3^)^         10.  {l-ixy\  11.  {i-ixy\ 

12.  Find  the  8th  term  (independently  of  the  rest)  in 
(a--xy. 

13.  Find  the  98th  term  in  {a-hy'\ 

14.  Find  the  5th  term  in  {a'-by\ 

15.  Find  the  middle  term  of  {a-{-xy\ 

16.  {x'  +  xyy.  17.   {a'-x'y,  is.  {a'  +  ¥y. 

308.  The  Binomial  Theorem  is  true  not  only  for  n  a  pos- 
itive integer,  bnt  for  n  negative  or  fractional.  But  the  dis- 
cussion in  this  case  is  not  sufficiently  elementary  for  this 
book. 


SCALES   OF  JS"OTATIOK.  269 

XLI.      Scales  of  Notation. 

309.  In  the  common  system  of  Arithmetic  numbers  are 
expressed  by  the  use  of  9  figures  called  digits,  and  one  ci- 
pher. This  is  effected,  we  know,  by  giving  to  each  digit  a 
local,  as  well  as  its  intrinsic,  value.  The  local  values  of  the 
figures  increase  in  a  tenfold  proportion  in  going  from  right 
to  left ;  in  other  words,  the  local  values  of  the  digits  pro- 
ceed according  to  the  poiuers  of  10  froin  right  to  left. 

Thus,  4296  may  be  expressed  by 

4000  +  200  +  90  +  6,  or  4XlO^  +  2XlO'^  +  9XlO  +  6. 

A  system  of  notation  is  called  a  scale.  In  the  common 
system  or  scale,  the  number  10  is  called  the  radix  or  base 
of  the  scale. 

310.  It  is  purely  conventional  that  10  should  be  the  radix ; 
and  therefore  there  may  be  any  number  of  different  scales, 
each  of  which  has  its  own  radix.  Notation  is  then  the 
method  of  expressing  numbers  by  means  of  a  series  of 
powers  of  some  one  fixed  number,  which  is  said  to  be  the 
dase  of  the  scale  in  which  the  numbers  are  expressed.  (We 
use  the  word  number  here  in  the  sense  of  whole  number,) 

If  the  digits  of  a  number  JSf,  of  n  digits  (including  0 
among  the  digits  for  convenience),  be  a^,  a^,  a^  ,  .  .  a^^^, 
reckoning  from  right  to  left,  and  r  be  the  radix,  N  may  be 
expressed  by  the  formula, 

Obs.  1. — It  will  be  noted  that  since  the  units  figure  does  not  contain 
r,  the  highest  power  of  r  will  be  one  less  than  the  number  of  figures  in 
the  number  expressed. 

Obs.  2.— In  any  scale  of  notation  every  digit  is  necessarily  less  than 
r,  therefore  r-1  is  the  greatest  digit,  and  r-1  expresses  the  number 
of  digits,  and  the  number  of  figures  used  in  any  scale  including  0  is 
equal  to  r. 

Arithmetical  Notation.  Scale.  Radix,  or  Base.  The  general  formula  for  ex- 
pressing numbers. 


^70  ELEMElsTTARY    ALGEBKA. 

311.  If  r=2  the  scale  is  called  the  Binary; 

r=3  Ternary ; 

r=4: Quaternary ; 

T=6   Quinary ; 

r=6 Senary ; 

&c.,  &c. 

T=  10  Denary ; 

^=11  Undenary ; 

r=zl2  Duodenary, 

The  digits,  including  the  cipher,  in  the 
Binary    scale  are  1,  0 ; 

Ternary ^,2,0; 

Quinary 1,  2,  3,  4,  0; 

&c.,  &c. 

Nonary  1,  2,  3,  4,  5,  6,  7,  8,  0; 

Denary   1,  2,  3,  4,  5,  6,  7,  8,  9,  0; 

but  in  the  duodenary  scale  we  must  have  two  additional 
characters  to  express  ten  and  eleven ;  we  therefore  put  t  for 
ten,  and  e  for  eleven. 

.•.  Duodenary  digits  are  1,  2,  3,  4,  5,  6,  7,  8,  9,  t,  e,0; 

also  undenary   1,  2,  3,  4,  5,  6,  7,  8,  9,  t,  0. 

All  numbers  are  supposed  to  be  expressed  in  the  common, 
or  denary  scale,  unless  otherwise  stated. 

312.  To  express  a  given  number  in  any  proposed  scale. 
Let  N  be  the  number,  and  r  the  radix  of  the  proposed 

scale. 

Then  if  «„,  a^,  a^,  &c.  be  the  unknown  digits, 

iV=:an-i^"~'  +  fl^^2^"~'+  .  .   ^-ay  +  ay  +  a^r+a^. 
If  now  N  be  divided  by  r,  the  remainder  is  a^. 
If  the  quotient  be  divided  by  r,  the  remainder  is  a^. 

Names  of  different  scales.      To   express   a  given   number  in  any  given  scale. 


SCALES   OF   2s"0TATI0i^. 


271 


303... 

...2 

.•.  1st  remainder  ^^=2 

50.... 

...3 

2d          "         a=^ 

8 

...2 

3d          "         «,=:2 

1 

...2 

4tli         "         a, =2 

If  the  second  quotient  be  divided  by  r,  the  remainder  is  a^ ; 
and  so  on,  until  there  is  no  further  quotient. 

Hence  the  repeated  divisions  of  the  given  number  N,  by 
the  radix  of  the  proposed  scale,  give  as  remainders  the  re- 
quired digits  of  the  number  in  the  proposed  scale. 

Ex.  1.  Express  1820  of  the  common  or  denary  scale,  in  a 
scale  whose  base  is  6. 

6  1820 

6 

6 

6 

6 

.-.  the  number  required  is  12232. 

This  is  easily  verified,  for 

1X6'+2X6'+2X6'  +  3X  6 +  2  =  1820. 

This  verification  gives  the  method  of  transforming  a 
number  from  any  other  scale  to  the  denary. 

By  the  method  of  division  given  above,  a  number  may  be 
transformed  from  miy  given  scale  to  any  other  of  which  the 
radix  is  given.  It  is  only  necessary  to  bear  in  mind  through- 
out the  process  that  the  radix  of  the  scale  of  the  given  num- 
bers is  not  10,  but  some  other  number.  Or  the  same  thing 
may  be  done  by  first  expressing  the  number  (as  by  verifica- 
tion above)  in  the  denary  scale,  and  then  proceeding  as  in 
Ex.  1. 

Ex.  2.  Transform  12232  from  the  senary  scale  to  the 
quaternary. 


To  transform  a  number  from  one  scale  to  another. 


4 

12232 

4 

-2035.. 

..0 

4 

305.. 

..3 

4 

44.. 

..1 

4 

11.. 

..0 

1.. 

..3 

272  ELEMENTARY    ALGEBRA. 

(Observe,  that  in  dividing  13  by  4,  12  does  not  mean  twelm,  but 
1  x6  +  2=:8 ;  so  also,  23  is  fifteen^  32  is  twenty^  and  so  on;  i.  e.  we  must 
convert  each  partial  dividend  to  the  denary  scale  as  we  proceed.) 


,  1st  remainder  ^^^^O 
2d  "  a,=:3 

3d  "  a,=l 

4tli  "  ^3=0 

5th  "  a,=0 

.'.  the  number  required  is  130130. 

This  number  transformed  to  the  denary  scale  is, 
1x4^  +  3x4*4-0x4^1x4^  +  3x4  +  0=1820. 

Ex.  3.  Transform  3256  from  a  scale  whose  radix  is  7,  to 
the  duodenary  scale. 

twelve   3256 

twelve     166 4    .*.  1st  remainder  aj,=: 4; 

11 1     .-.  2d  "  a=l] 

.'.  the  number  required  is  814. 

(Observe  in  this  division  that  33  is  twenty-three,  and  the  remainder, 
eleven,  is  multiplied  by  7  and  added  to  the  next  figure,  5,  giving  eighty- 
two  for  the  next  partial  dividend,  &c.) 


Examples — 71. 


t.  Express  the  common  number  300  in  the  scales  of  2,  3, 
4,  5,  6. 

2.  Express  10000  in  the  scales  of  7,  8,  9,  11,  12. 

3.  Express  a  million  in  the  duodenary  scale. 


SCALES   OF   IS'OTATION". 


273 


4.  Transform  27z^  and  7007  from  the  undenary  to  the  oc- 
tenary  scale. 

5.  If  the  number  803  is  expressed  by  30203,  show  that 
the  new  scale  is  the  quaternary. 

6.  The  number  95  is  expressed  in  a  different  scale  by  137 : 
find  the  base  of  this  scale. 

313.  The  common  processes  of  Arithmetic  are  all  carried 
on  with  numbers  expressed  in  any  one  of  these  scales  as 
with  ordinary  numbers,  observing  that  when  we  have  to  find 
what  numbers  we  are  to  carry  in  Addition,  &c.,  we  must 
not  divide  by  ten,  but  by  the  base  of  the  scale  in  which  the 
numbers  are  expressed. 


Ex.1. 


Addition, 

r^A: 

r=zl 

32123 

65432 

21003 

54321 

33012 

43210 

22033 

1444 

31102 

65001 

332011   326041 


Subtraction, 
r=3  r=12 

7^^348 
he^t^ 


201210 
102221 

21212 


1^864 


Ex.  2.    Multiply  the  numbers  1049  and  1^5  together  in  the 
duodenary  scale. 

1049 
lg5 
51^9 
e443 
1049 
202329 -duodenary. 

.*.  the  product  is 

202329=2X12^  +  2X12' +  3x12^+2x12  +  9 

=  501585~denary. 
12^ 


274  ELEMEKTAKY     ALGEBRA. 

Ex.  3.    Divide  234431  by  414  (quinary),  and  extract  the 
square  root  of  122112  (senary). 


234431 
41 

414)234431(310 
'  2302. 

122112(252 
4 

122112 
44 

234340 

423 
414 

45)421 
401 

122024 

41 

542)2012 
1524 

• 

44 

814.  To  find  the  greatest  and  least  numders  expressed  hy  a 
give7i  number  of  figures  in  arty  proposed  scale. 

Let  r  be  the  base  of  the  scale,  and  n  the  number  of  digits ; 
then  the  number  will  be  greatest  when  every  digit  is  as  great 
as  it  can  be,  that  is,  =r— 1.     Thus  the  number  will  be 

(r-l)r"-^+(r-l)r"-2+    ....   +(r-l)r'  +  (r~l)r+r-l; 

or,  (r-1)  (r^Hr"-'+  ....   ^r^^r-{-l). 

But  the  quantity  in  the  second  parenthesis  is  the  sum  of 
the  terms  of  a  geometrical  progression,  of  which  the  first 
term  is  r"~^,  the  ratio  r,  and  the  last  term  1.    This  is  equal  to 

7-^1 

— - .    We  have  then  for  our  greatest  number, 

(r—1) -:     or,  r"— 1. 

Again,  the  number  will  be  least  when  the  digit  on  the  left 
IS  1,  and  all  the  other  figures  0,  in  which  case  it  will  be  equal 
to  r^\ 

Ex.  1.  In  the  denary  scale  the  greatest  number  of  3  fig- 
ures is  10'- 1:^999;  and  the  least  is  10',  or  100. 

To  find  the    greatest    and   least    numbers  expressed    by  a  given    number  of 
figures  in  any  proposed  scale. 


LOGAKITHMS.  275 

Ex.  2.  In  the  senary  scale  the  greatest  number  of  3  fig- 
ures is  555  =  6^  —  1  =  215,  denary;  and  the  least  number  of 
3  figures  is  100  =  6^=36,  denary. 


Examples— 72. 

1.  Extract  the  square  root  of  33224  in  the  scale  of  six. 

2.  Show  that  144  is  a  perfect  square,  in  any  scale  whose 
radix  is  greater  than  four. 

3.  Show  that  12345654321  is  divisible  by  12321  in  any  scale 
greater  than  six. 

4.  Multiply  the  common  numbers  64  and  33  in  the  binaiy 
and  quaternary  scales,  and  transform  each  result  to  the 
other  scale. 

5.  Divide  51117344  by  675  (octenary),  37542627  by  42?f 
(undenary),  and  29^96580  by  "2U^  (duodenary). 

6.  Extract  the  square  roots  of  25400544  (senary),  47610370 
(nonary),  and  32^75721  (duodenary). 

7.  Express  in  common  numbers  the  greatest  and  least 
that  can  be  formed  with  four  figures  in  the  scales  of  6,  7, 
and  8. 

8.  Show  that  1331  is  a  perfect  cube  in  any  scale  of  nota- 
tion whose  radix  is  greater  than  three. 


XLII.         LOGAEITHMS. 

315.  A  geometrical  progression  whose  first  term  is  1  and 
ratio  any  number,  as  a,  may  be  written 

a\  a',  a",  a\  a\  a',  a',  a\  a^  &c.,  a^ ; 

and  the  indices  form  the  arithmetical  progression 

0,  1,  2,  3,  4,  5,  6,  7,  8 n. 


27G  elemein^tary  algebra. 

In  this  A.  p.  each  term  measures  the  order  of  the  ratio  of 
the  corresponding  term  in  the  geometrical  progression  to  1.* 
Hence  these  indices  are  called  the  measures  of  the  ratios  of 
the  numbers  in  G.  P.  to  1,  or  the  Logarithms  of  these 
numbers. 

316.  From  the  rules  established  in  the  earlier  chapters,  we 
know  that  to  multiply  or  divide  any  two  terms  in  the  first 
series  we  have  only  to  add  or  subtract  their  indices — i,  e,,  the 
corresponding  terms  in  the  second  series;  also,  to  raise  a 
number  of  the  first  series  to  a  given  power,  we  multiply  its 
index  or  corresponding  term  in  the  second  series  by  the  in- 
dex of  the  power ;  also,  to  extract  a  given  root  of  any  number 
in  the  G.  p.,  we  divide  its  index  or  corresponding  term  in  the 
A.  p.  by  the  index  of  the  root. 

317.  It  is  evident  from  the  above  that  if  a  geometrical 
progression  can  be  formed  which  shall  represent  with  a  suffi- 
ciently close  approximation  all  numbers  from  1  to  10,000..  and 
the  terms  of  the  arithmetical  progression  corresponding  to 
this  G.  p.,  in  the  same  manner  as  in  Art.  315,  be  calculated, 
and  both  series  be  recorded  in  a  table,  that  much  trouble 
may  be  saved  in  arithmetical  computation  by  operating 
solely  on  the  terms  of  the  A.  p.,  and  finding  from  the  table 
the  numbers  of  the  g.  p.  corresponding  to  the  results. 

318.  Such  tables  have  been  calculated,  and  are  called 
TaUes  of  Logarithms,  To  see  how  this  may  be  efiected, 
let  a=10  m.  the  system  (Art.  315) ;  we  have  then 

lo^  10^  lo^  lo^  lo^  lo^  lo^  lo^  lo^  lo^  &c.,  (i) 

and    0,     1,     2,      3,      4,      5,      6,      7,      8,      9,  (2) 

are  the  logarithms  of  the  corresponding  terms  of  the  first 
series ;  that  is,  in  a  system  of  logarithms  whose  base  is  10, 

*  The  ratio  of  a?  -.1  is  the  duplicate  ratio  of  a:  1. 

The  ratio  of  a^ :  1  is  the  triplicate  ratio  oi  a\l. 

And  the  ratio  of  a" :  1  is  called  n  times  the  ratio  of  a ;  1. 
Thus  the  indices  Tneasure  tJie  ratios. 


LOGAEITHMS.  277 

O=log.  10"  or  log.  1; 
l=log.  10^  or  log.  10; 
2=log.  10' orlog.  100; 
3= log.  10' or  log.  1000; 
4=log.  10*  or  log.  10,000 ; 
5=log.  10' or  log.  100,000; 
&c.,  &c.,  &c. 

It  is  manifest  now  that  the  arithmetical  mean  between 
any  two  terms  of  the  series  (2)  will  be  the  logarithm  of  the 
geometrical  mean  between  the  two  corresponding  terms  of 
the  series  (1). 

The  arithmetical  mean  of  0  and  1  is  —^r—  =  .5. 

The  geometrical  mean  between  1  and  10  is 

^/fxlO  ==3.16227  +  ; 

and  therefore  .5  =  the  logarithm  of  3.16227. 

The  arithmetical  mean  of  .5  and  1  is  .75. 

The  geometrical  mean  of  10  and  3.16227  is 

Vl0x3.162"27= 5.62341  +  ; 
whence  .75  =  the  logarithm  of  5.62341. 

The  arithmetical  mean  of  1  and  2  is  1.5. 

The  geometrical  mean  of  10  and  100  is  31.62277+; 
whence  1.5  =  the  logarithm  of  31.62277. 

And  by  repeating  this  process,  with  immense  labor,  the 
inventor  of  logarithms,  N"apier  (a.  d.  1618),  and  his  suc- 
cessor in  these  calculations,  Briggs  (a.  d.  1624),  calculated 
tables  of  logarithms  of  natural  numbers  from  1  to  100,000, 
But  (he  labor  of  calculating  logarithms  is  much  diminished 


278  ELEMEl^TARY    ALGEBEA. 

by  the  use  of  series  which  cannot  find  place  in  an  ele- 
mentary work  like  the  present ;  besides,  as  will  be  seen,  the 
chief  labor  is  with  the  prime  numbers. 

319.  These  logarithms  in  the  common  table  of  which  10  is 
the  base,  are  the  indices  (entire  or  fractional)  of  the  powers 
to  which  10  is  to  be  raised  to  obtain  all  natural  numbers  ap- 
proximately. Thus,  .30103  is  the  logarithm  of  2,  means  that 
2Q.3oio3__2.     And  this  may  be  yerified  by  developing  10-^*^^°^ 

.30103 

~  (1  +  9)ioooob  by  the  binomial  formula. 

820.  10  is  the  most  convenient  base,  but  any  positive 
number  except  1  may  be  taken  as  the  base.  E"apier  took 
2.71828  as  his  base.  In  general,  by  taking  any  positive 
number  (except  unity)  for  a  base,  we  may  express  any  posi- 
tive number  as  some  power  of  it.  And  thus  logarithms  may 
be  defined  to  be  the  indices  of  the  powers  {entire  or  frac- 
tional) to  which  ice  raise  a  fixed  number,  called  the  hase,  to 
oUain  the  series  of  natural  numhers.  Each  logarithm  is  the 
representative  of  its  corresponding  natural  number. 

321.  In  the  common  system,  of  which  10  is  the  base,  it  is 
clear  that  the  logarithm  of  every  number  between  1  and  10 
is  a  decimal  fraction ;  that  of  everv  number  between  10  and 
100  is  1  with  a  decimal  fraction  annexed;  that  of  any 
number  between  100  and  1000  will  be  2  with  a  decimal  frac- 
tion annexed,  &c.  The  integral  part  of  a  logarithm  is 
called  the  characteristic  of  the  logarithm ;  and  the  decimal 
part  is  called  the  Mantissa,  or  "  handful.'^  Thus  0  is  the 
characteristic  of  the  logarithms  of  numbers  between  1  and 
10 ;  1  is  the  characteristic  of  the  logarithms  of  all  numbers 
between  10  and  100 ;  2  that  of  the  logarithms  of  all  numbers 
between  100  and  1000,  &c.  And  in  general,  the  character- 
istic of  the  logarithm  of  any  number  is  alioays  less  ly  unity 
than  the  number  of  figures  in  the  given  number. 

322.  Tables  of  logarithms,  arranged  in  convenient  form, 
are  usually  given  in  books  on  Trigonometry,  and  with  them 


LOGARITHMS. 


279 


explanations  of  the  mode  of  finding  in  the  table  the  loga- 
rithms corresponding  to  a  given  number,  or  the  number 
corresponding  to  a  given  logarithm.  The  table  below  is  a 
portion  of  such  a  table  of  logarithms. 


Logaritlims^  to  hose  10,  of  all  Prime  Numbers  f 

yo7n 

1  to  100. 

No. 

Logarithms. 

No. 

Logarithms. 

No. 

Logarithms. 

No. 

Logarithms. 

2 

0.3010300 

19 

1.2787536 

43 

1.6334685 

71 

1.8512583 

3 

0.4771213 

23 

1.3617278 

47 

1.6720979 

73 

1.8633229 

7 

0.8450980 

29 

1.4623980 

53 

1.7242759 

79 

1.8976271 

11 

1.0413927 

31 

1.4913617 

59 

1.7708520 

83 

1.9190781 

13 

1.1139434 

37 

1.5682017 

61 

1.7853298 

89 

1.9493900 

17 

1.2304489 

41 

1.6127839 

67 

1.8260748 

97 

1.9867717 

323.  We  will  now  show  more  fully  the  properties  of  loga- 
rithms, which  render  them  so  useful  in  diminishing  the 
labor  of  arithmetical  calculations. 

324.  In  the  same  system,  the  sum  of  the  logarithms  of  two 
numbers  is  the  logarithm  of  their  product ;  and  the  difference 
of  the  logarithms  of  two  numlers  is  the  logarithm  of  their 
quotient. 

Let  m  and  n  be  the  two  numbers ;   lei  x  =  log.  m,  and 

y  =  log.  n;  let  a  be  the  base  of  the  system;   then  a''=7n, 

on 
and  a^—n\  hence  a'^'^—mn,  and  a'^^——\  or  x-^y  is  the 

m 
log.  mn,  and  x—y  is  log.  — ;   that  is,  log.  m  +  log.  ^  =:log. 

n 

7)1/ 

mn  ;  and  log.  m  —  log.  n  —  log  — . 

n 

Ex.  1.  Log.  6=log.  2+log.  3  =  .3010300  +  4771213 

=.7781513. 
Ex.  2.  Log.  mnp=log.  mn  +  log. p=log.  m  +  log.  n-\-log,p. 
Ex.  3.  Log.  5=log.  10-log.  2=:l-log.  2=.6989700. 
Ex.  4.  Log.  |=:log.  7-log.  5=.1461280. 


280  ELEMENTARY    ALGEBRA. 

Ex.  5.  Log.  .07=log.  T^=log.  7-log.  100:=:. 8450980 -2, 

which  is  written  thus,  2.8450980;  it  being  understood  that 
in  this  position  of  the  negative  sign  it  belongs  only  to  the 
characteristic  2,  and  not  to  the  mantissa,  which  is  still 
positive. 

Ex.  6.  Log.  ^\=.4771213-1.9867717  = -1.5096504, 
which  maybe  written  thus:  -2 +  (1 -.5096504)  =2.4903496. 

S25.  If  the  logarithn  of  a  number  be  multiplied  by  m,  the 
product  is  the  logarithm  of  that  number  raised  to  the  m.th 
fower. 

Let  iV^be  the  number  whose  logarithm  is  x\  then  a^=N\ 
therefore  a'^'^=N'^\  that  is,  mx  is  the  log.  of  N''^\  or  log. 
N'^—mx^m  log.  N, 

Ex.  1.  Log.  (13)^=5xlog.  13  =  5X1.1139434=5.5697170. 

Ex.  2.  Log.  b^—%j  log.  b. 

Ex.  3.  Log.  4= log.  2=^  =  2  log.  2  =  .6020600. 

Ex.  4.  Log.  (a''-xy=^2  log.  (a  +  a;)+2  log.  {^a-x), 

Ex.  5.  Log.  (a"'Z>"c^. .  .)=m  log.  a^-n  log.  b-Vp  log.  c-f-. . . 

326.  If  the  logarithm  of  a  number  be  divided  by  m,  the 
quotient  is  the  logarithm  of  the  mth  root  of  that  numMr 

Let  a;=log.  iV,  or  a'^—N^ 

X     log.  N 


then  arfi  —  N^^  or  log.  iV^»n= 


m        m 


^     ^  r        K^    %-^     .6989700      ,^,^,^^ 

Ex.1.  Log.  5*=—^= J- — =.1747425. 

Ex.  2.  Log.  y -|-=—  log.  a-~  log.  b, 

Ex.  3.  Log.  Va^—x^=i  log.  {a  +  x)-{-i  log.  (a—x). 


LOGAEITHMS  281 

Ex.  4.    Given  log.  128=2.1072100  to  extract  the  7th  root 
of  128. 
Log.  VlM=\  log.  128=1  (2.1072100)  =.3010300=log.  2. 

.•.V'l28=2. 

Ex.  5.   Log.  -VTt^^  log.  1-1  log.  71= -I  log.  71 
=4-X  ~L8512583=|  (-2+(l-.8512583))=4-  (2.1487417). 
]^ow  in  order  to  diyide  this  log.  by  7,  we  place  it  under 
the  form  7  +  5.1487147,  so  that  the  negative  characteristic 
may  become  a  multiple  of  7 ;  then 

4-  ( 7  +  5.1487147) = f.7355306. 

327.  We  can  now  see  how  the  work  of  computing  a  table 
of  logarithms  is  facilitated  by  the  application  of  the  above 
properties  of  logarithms.  For  the  logarithms  of  the  com- 
posite numbers  are  all  found  by  adding  together  the  loga- 
rithms of  their  prime  factors. 

328.  While  we  are  at  liberty  to  take  any  number  except  1 
as  the  base  of  a  system  of  logarithms,  we  can  now  under- 
stand the  great  advantages  of  the  system  which  has  10  as  a 
base ;  that  is,  the  advantages  of  having  the  base  of  the  scale 
of  notation  the  same  as  the  base  of  the  system  of  logarithms. 

For,  1st,  the  characteristic  of  the  logarithm  of  any  whole 
number  is  always  one  less  than  the  number  of  figures  in  the 
given  number.  Hence  when  the  number  of  figures  is  given 
we  know  the  characteristic ;  and  when  the  characteristic  of 
the  logarithm  is  given,  we  know  the  number  of  figures  in  the 
required  number. 

2d.  By  every  multiplication  or  division  of  a  number  by 
10,  the  characteristic  of  its  logarithm  is  increased  or  dimin- 
ished by  unity. 

For  log.  1280= log.  (128X10) 

=log.  128  +  log.  10=log.  128  +  1, 


282  ELEMENTARY    ALGEBRA. 

log.  128u0=log.  128  +  log.  100=:log.  128+2, 
log.  128000=  log.  128+log.  1000=log.  128  +  3, 
log.  12.8=.log.  (128^10)=log.  128-log.  10=log.  128-1, 
log.  1.28=log.  (128 -f- 100)=  log.  128-log.  100=:log.  128-2, 
log.  .128=log.  128-log.  1000=log.  128-3,  &c. 
From  the  tables  of  logarithms,  log.  128=2.1072100. 
Therefore,  log.  1280      =3.1072100, 

log.  12800  =4.1072100, 

log.  128000  =5.1072100, 

log.  1280000=6.1072100, 

log.  12.8        =1.1072100, 

log.  1.28        =0.1072100, 

log.  .128        =1.1072100, 

log.  .0128      =2.1072100. 

We*  observe  that  the  logarithms  of  all  numbers  which 
contain  the  same  significant  figures,  arranged  in  the  same 
manner,  have  the  same  mantissa,  or  decimal  parts ;  that  is, 
the  mantissa  of  the  logarithm  remains  the  same  however  we 
may  change  the  corresponding  number,  by  annexing  ciphers 
or  by  inserting  a  decimal  point  in  it,  changing  the  position 
of  its  decimal  point  to  the  right  or  left. 

Moreover,  the  logarithm  of  a  decimal  fraction  has  a  nega- 
tive characteristic  greater  by  unity  than  the  number  of  O's 
between  the  decimal  point  and  the  first  significant  figure  of 
the  number. 

329.  To  find  a  fourth  proportional  to  three  given  numbers, 
using  logarithms. 

Let  the  numbers  be  a,  b,  and  c;  let  x  =  required  fourth 

be 
proportional.    Then  a  :  b=c:x;  .*.  x=—. 


LOGARITHMS.  283 

Therefore  log.  x=log,  J  +  log.  c— log.  a.     Hence  tlie  rule  : 

From  the  sum  of  the  logarithms  of  the  second  and  third 
terms  subtract  the  logarithm  of  the  first  term :  the  remainder 
will  be  the  logarithm  of  the  fourth  proportional  The  fourth 
proportional  may  then  be  found  from  the  tables. 

Note. — In  the  following  examples  use  the  table  of  logarithms  given 
in  Art.  322. 

Examples — 73. 

1.  Find  the  logarithms  of  8,  9,  12,  20,  25,  60. 

2.  Find  the  logarithms  of  |,  i,  f,  .03,  ■^\,  .0033. 

3.  Eequired  the  logarithms  of  168,  1.04,  and  3690. 

4.  Given  the  logarithms  of  3  and  7 :  find  the  logarithm  of 
14700. 

5.  Find  the  logarithm  of  83349,  from  the  logarithms  of 
3  and  .21. 

6.  Determine  the  logarithms  of  V^-^%  and  \/l.625,  by 
means  of  those  of  2,  3,  5,  and  13. 

7.  Find  a  fourth  proportional  to  the  quantities  1.3,  .0104, 
and  2.375,  by  logarithms. 

8.  Find  by  means  of  logarithms  the  number  of  figures  in 
the  results  of  the  involutions  of  2^°  and  3'^ 

9.  Find  the  logarithm  of  V^J"  X  V^f"  X  Vf. 


ANSWERS   TO    EXAMPLES. 


Examples— 1. 

I.  22,        2.  26.        3.  89.        4.  564.  5.  274.        6.  10. 

7.  6.       8.  6.      9.  34.       10.  39.       11.  6.       12.  5.       13.  9. 

14.  5.       16.  x  +  a  +  b,       17.  x-}-x-^a  +  x  +  a  +  b.       18.  x—a. 
19.  aS'-^.  20.  x~a  +  b,  21.  2:  +  a:+2+2:-f2+3. 

22.  4xlO  +  5+yV  23.  100a;+10^  +  ^,  100;^ +  10y +  .'?;. 

24.  X7i  +  bn.         25.  lla;+5.  26.  ~.        27.  — .       "^ 


^*  'a;      a;4-3' 


iV— r 

28.     — y^. 


Examples — 2. 

1.  55. 

2.  81. 

3.  94.        4.  8. 

5.  27. 

6.  81 

7.  13. 

8.  11. 

9.  31.        10.  15. 

•11.  10. 

12.  3 

3.  3. 

U.  127. 

answers  to  examples.  285 

Examples — 4. 

1.  5.         2.   16.         3.  9.  4.  224.         5.  459.  6.  7. 

7.  74.  8.  12.  9.  8.  10.  238.  11.  420.  12.  144. 
13.  43.        14.  15. 

Examples — 5. 

1.  2a+2b.         2.  2a,        3.  2a-2b.       4.  2a,        5.  2a+2^. 

6.  2+m  +  n.  7.  7m— 1.  8.  4:xy-h4:X.  9,  p—q-^-S, 
10.  eah-bc,  11.  15a-9^.  12.  dx'-df, 
13.  9«  +  95+9c.  14.  4:z;+2y  +  4^.  15.  a—b. 
16.  3a;-3a-2Z>.  17.  2a+2b.  18.  a  +  ^  +  c. 
19.  --2«  +  2J  +  2^.  20.  2a;'  -  2x'  -  8x  +  10. 
21.  5^*  + 4a;' +  3^' +  2^- 9.  22.  4:a'  +  2a'h-4.a¥+b'-n\ 
23.  a'x  +  da\  24.  6aJ-9a'a;  +  7aa;'  +  ^x^  25.  5a;^ 
26.  10a;^  +  83/'4-12a;+12. 

Examples — 6. 

1.  3^  +  45.                  2.  4:a+2c,                   3.  a+5b  +  4:C  +  d. 

4.  2.T^-2a;--4.         5.  3a;* -a;' -14a; +18.        6.  x'-ax  +  2a\ 

7.  -5a;y-5a;^4-2i/'  +  7/^.        8.   3a;'  +  13a:y-16a;2;-y'-13y;2. 
9.  2a'-6a'b  +  6ab'-2b\         10.  3a;'' 4- 4a; +  16,  a;H8a;'. 


286  eleme^'tary  algebra. 

Examples— 7. 

1.  4.a-4:X.  2.  4:a'-4.a'c.  3.  x'-3y'-3z\ 

4.  2ax''  +  2by'+2cz\  5.  a'-dd'+3c\  6.  2ab-i-U\ 

7.  0.         8.  -3x-y-h4.z.  9.  8a;-8.  10.  -4:C  +  4:d. 

Examples — 8. 
1.  {a-h+c)x'-{I)-c-{-d)x^-{c  +  d+e)x,       2.  2{ax-dy). 

3.  {a-\-d)x^—{a—5b)xy-}-{a—c)y\ 

4.  2{ax-\-cy),  2b{x+y). 

5.  — («— 5J):r+(2(^+3Z>  +  c)y,  (aj— 45  — c)^+(«^— 35— 2^)?/, 
(5— c):c+(3a— c)y. 

6.  {6a—b)x—{2a-db'-6c)y,       —{a-^-c)x+{a—b+2c)y, 
(4.a-b-c)x-{a-2b-7c)y. 

Examples — 9. 

1.  abx^y%  —mnx^,  2a'cx'^y,  ab'^c^,  d^bc^^  -~^y« 

2.  x^—x'y^-xy'',    —a^x-\-d^x^—aQ^^     —abx^-\-(^bx^—ab'^x^ 
x'y-3x^y^-\'3xY-xy\ 

3.  20" ■\-1ab-\-W,2ac-bc-Ud+3bd. 

4.  ^x'  +  lZxy  +  Qy^    Mb'-ab'-Ub*. 

5.  x'+Qx'-rlx-^e,    x'-ex'-^Ux-G. 

6.  a*+a'-2a'+3a-l,    a*-a'-8a'4-a+l. 


^  VNIYER" 

AKS^RS  TO   ;gXAMPLES.  287 

SAUfQ    : 

7.  Six' -if.  8.  a''\-S2b\  9.  x'-ia'x+da'. 

10.  27a'  +  ¥  +  S-lSab,  11.  x'-y'+z'  +  Sxyz 

12.  a'-l.      13.  a'-8Z>'-27c'-18a^c.      14.  a'-h2aW-{-b\ 

15.  a;'— (a+c)cc''  +  («^c  +  ^)^ — be; 

x*—(a''—b+c)x'  +  a{b  +  c)x—bc 

16.  1  — (a— l)a;— (a— d  +  l)^'  +  (fl^  +  Z>-c)^'  — (^>+c)2;*  +  c:?;'. 


Examples— 10. 

1.  a'-2ax+x\  l-^ix^+4:x\  4a*  +  12a'  +  9> 
9a;''-24a;y^-lG?/^ 

2.  9  +  12:^:  +  4a;^         4:i;^-12:ry  +  9/,  a*-6a'c?;+9aV, 
b'x*'-2bcx'y  +  c'xy. 

3.  4^^^-!,  9aV-Z>^  a;*-l. 

4.  x'+4.x  +  d,  x'+3x'-4:,  a'b'-qb-Q,  4.aV-Sabx+db\ 

5.  x'-5aV  +  4:a\  6.  mV-13mV2;y  +  36^y. 
7.  4^1  8.  x'+4.y\  4.a*-6a'b'-}-b\ 
9.  a'  +  2r/^»+^>^-c^  «'^^>''+2^c  +  c^  a'-b'-2bc—c\ 

10.  «'-2aJ  +  ^>''-c^  -a^  +  2«5-^>•'+c^  -^'+Z>'-2Z^c  +  c'. 

11.  4ta'-b'  +  6bc-9c',  -4:a'  +  12ac-hb'-9c\ 

12.  4a^-Z>''-6^^c-9c^  -4a'-{-4.ab-b'  +  dc\ 

13.  a=+2ac+c'-J'-2^>^-^^      a''  +  2ac?+6Z'-^'-2^^c-c', 


288  ELEMENTARY    ALGEBRA. 

+  ^ab  -  ib%     a'  +  6ac  +  9c'-  W  +  Ud-  d\  . 

Examples — 11. 
I.  5a;'.  2.   -3a^  3.  ^xy.  4.   -MhW 

5.  ^a'Wy\  6.  x^-'^x^L  7.  -aM-4a-5. 

8.  x^-Zxy^A.f,  9.  5«^P+a^-4. 

10.  15a'^'-12«JH9aJc'-5c\  11.  x-L         12.  a;- 8. 

13.  a;^  +  a;  +  3.  14.   3ii;'^-2a;  +  4.  15.   3:z;''  +  2:z;  +  l. 

16.  x^-Zx^l.  17.  2;^  +  a;*  +  2;'  +  ^'+i?;+l. 

18.  a^-\-ab-l\  19.  a;^  +  3^>+9^^''+272/'. 

20.  x^-x'y  +  xy\      21.  i?;*+ii^>  +  i^y +  :2^^'+y'. 

22.  a*-2a^Z>  +  46^^Z>^-8t^Z>'  +  16^\ 

23.  2a'-Mb  +  l^aV-mh\  24.  a;''+^y+^'. 
25.  a;'  +  2:r2/  +  3/.  26.  x^^^x-{-2.  27.  a;'-3:c-l. 
28.  ic'-5a;+6.  29.  :z:'-4a;  +  8.  30.  a;''  +  5:i;+6. 
31.  x-c,                      32.  a;'-j9a;  +  5'. 

33.  y'-{m-l)y^-{m-n-l)y''-{m-l)y  +  l. 

34.  a'  +  Z>'  +  c'  +  fl^&-flJc  +  Z>c,  a'  +  ^'+c'  +  a^  +  «c-2><:. 

35.  a—ax-{-ax^  —  ax^-{-  -^ — ,  l  +  5a:+15a:'  +  45a;'  +  - — ^ 

l-{-x  l—6x 

36.  l  +  2a:  +  3a;^  +  4a;H  ,     ^        ,. 

1— 2a;4-ar 


AKSWlrKS  TO   EXAMPLES.  289 

Examples — 12. 

1.  v.—x,a^-\-a^x^ a^x^  +  ax^  +  a?^  w'-a^x  +  a^x^-d^x^  +  ax*' — x^, 

2.  3rc  +  l,  5:c-l,  2ic-3.  3.  Zmn-h,  hn^-n^, 

4.  l-2z  +  4^',   92:'  +  3^  +  l,  l--2a;+4^'-8a;'. 

5.  ^'  +  3a;>+9:?;2/^  +  27^',  a*-2^^Z>  +  4a'^Z>'-8a^^^  +  16^>*, 
x''-x'Y  +  xhf-xY  +  x'y^-y'\ 

6.  ia'-iaZ^^-^^  x\f-xYz-\-xyz'-z\ 

7.  «+Z>  +  c,  a+^— ^. 

8.  (o^  +  y)''— (a;  +  ^)2;  +  2;^=a;''  +  2a:?/+^''— ir^;— y^  +  2;^ 

Examples— 13. 

1.  (l-2:r)  (l+2.r),  (a-32:)  (<^+3a;),  (3m-27^)  (37?i+2^), 
a;X5a-2)  (5a  +  2),  xy{4:X-6ij)  (4a;  +  5?/). 

2.  (^+^)  (a;^_:r^  +  2/^),  (^-7/)  (x'+xtj  +  f),  (l  +  xy) 
(l-xy+xY),  (^-1)  (^+1)  (^'  +  1).  ^y((^y-^")  {cty^-x"), 
2«Z''c(<^-2c)  («+2c). 

3.  x\bx-a)  {bx-\-a),  a\a-3I)')  {a  +  SI?'),  (2x-S) 
(4^^ +  6.'?;+ 9),  (a-2h)  (a'  +  2ad  +  4.b'),  x'y{a  +  dy) 
{a''-3ay  +  9y"), 

4.  {x+2){x'-2x'-\-4.af-Sx+16),  x'{a  +  dx)  {a' -3ax+9x'), 
{2x'+f)  {4.x'-2xY+/),  {ab'+c')  {ab'-c")  {a'h'^-c^ 
abc(a-\-cy. 


290  ELEMEl!fTARY  ALGEBKA. 

5.  {3x-l)  (3a;  +  l)  (dx'  +  l),      {x^2)  (x+2)  {x'  +  2x  +  4:) 
{x'-2x  +  4.),  x\x-'b)\  x\x-ay  (x+a)\ 

6.  (4a;— 5) (2a; +  1),  {a+^h)  {a-b),  7{x-y){x+v). 

7.  {x-yy{x+7/)%  {c  +  a-I)){c-a  +  I?),  Sal?, 

8.  (x+yY,  mn{m--7i),  61){a—b). 

9.  2{x+y){4.x-y),  2{x-y){4.y-x),  ^y(x-\-y), 
10.  {a-V'b){a'  +  ah^l)''),  (a-by,  0. 

Examples— 14. 

1.  (x  +  l)  {x  +  6),    {x  +  4:)    {x  +  5),    {x-2)  {x-d),    {x-3) 
(x-5),  (a;+l)(a;+7),  {x^l){x-9), 

2.  {x  +  3)  {x-2),    {x-3)  {x  +  2),    {x-d)  {x+1),    {x+5) 
{x-3),  {x+S)  {x-1),  {x-9)  {x+1). 

3.  {2x+3){2x  +  l),  (4a;  +  l)(a;  +  3),  (4a;-l)(a;+3),  (2a;-3) 
(2a;+l),  (3a;-2)  (a;+2),  (3a;  +  4)  (2a;-l). 

4.  (4a;  +  l)(3a;-2),  2(6a;~l)(a;-l),  (4a;  +  l)(3a;-l),  (a;  +  4) 
{x-3),  (3a;-5)(a;+l). 

5.  a\x—a)  {x—2a),  a{a—Sx)  (a  +  2a;),  db{Za—2li)  (a+5), 
(2«+a;)(2a-a;)(3a'+a;'). 

6.  xy{2x'\-y)    {x-\-2y),     3y\3x-+2y)    {x-y),    a\3ax-l) 
{2ax-\-l),  x\2b-dx){db+x). 


ANSWERS  TO   EXAMPLES.  291 


Examples — 15. 


1.  2x'{a  +  xy.                2.  x\a+x)\          .  3.  ab{a-l))\ 

I.  2(^-1).                    5.  x\x^l),  6.  2(a;+a), 

7.  a\x-\-l),         8.  3(6?:?;  +  2).         9.  x-l,  10.  iz;  +  5. 

11.  a;-10.         12.  x'-x+l,        13.  a;  +  3y. 

Examples— 16. 

1.  3x-2.        2.  2a;  +  3.        3.  dx+6,        4.  8a;'' + 14a;- 15. 
5.  4a;-5.        6.  x'  +  2x-d. 

Examples — 17. 

1.   3:r~2.         2.   dx-2,         3.   2(:rH2:r+l).         4.  y-2. 
5.  :^;-2«.  6.  x+d.  7.  3{:?;-|-3).  8    x' i-y'. 

9.'a{a-{-h).  10.  a{a'-V).  11.  a;^~2:?;^-i'y'. 

12.  a;'+4a;  +  4 

Examples — 18. 

1.  12aWc,  S6x^y%  ax^y—axy\  ab^—ad^. 

2.  120a*b%  10a'b\  ISOOaV. 


?92  ELEMENTARY   ALGEBRA. 

3.  6{a'-b'),  12a{a'-l),  120xy{x'-y'), 

4.  24.a'b\a'-I)'),  d6xf{x'-y'). 

5.  (^  +  l)(^H-3)(2;-4).  6.  {x  +  2){x  +  4:){x'  +  3x  +  l), 

7.  x{2x  +  l){dx-l){4.x  +  3), 

8.  {x'-5x+6){x-l){x-4:). 

9.  (^'  +  3a;  +  2)(:?:-3)(:r+5). 

10.  {x'+x  +  l){x'+l){x  +  l){x-l), 

11.  36a^Z^V.  12.  120{a  +  by{a-hy, 
13.  24(a-^)(a'+Z>^).                   ^      14.  106ab'{a+I)){a-'b), 
15.  ^'-1.                        16.  0:^-1.  17.  x''-l. 
18.  (.T  +  1)  (:?:+2)(^  +  3). 

Examples — 19. 


1.  3:;^f  -^.        2.  4ac+7r.        3.  2aH-  — .         4.  2x--f- 
3.  ^4.  _A^.        6.  2a; ^.        7.  a;^  +  3aa;  +  3a^+-  ^^^' 


a; +  3*  *  x—3'  *  '  x—2a 

8.  ^-1 — ,        \.  9.  x'+x'i-x-hl-^-^. 

x—x  +  l  x—\ 


ANSWEES   TO    EXAMPLES.  293 


a-^b  '  '•    3{a  +  by  x'  +  l 


Examples — 20. 


^      2a^x  ^^    a-i-b  „    a+b  .        2ax 

"%"•                   ~2b~'  ~^:-b'  H^-^'y"" 

h[a—b)                 '        a—b  '  '  x  +  6' 

c     cc  +  T  rt   CC4-3  ^^    i?;+5  ^^    x—b 

CK— 5                    ic— 7  x-hc  x  +  G 


iA      3:r— 4              -„    x-\-a—b—c  ^.  x  +  3 

IZ.     —. ^.  Id.  — — 1 •  14. 


4^;— 3'  '    x-\-b—a—c  '     x^ —  2x-\-b 


X~'~0  ^  g^  X -IT  u  ^  jy  X 


•O-  '_    m-Ul- 


.-r:--!     x^  +  a'     a'+g'y  +  y      a:'- 5a:     a-5 
"a  "'       z'    '         a^  +  J'     '      x  +  b  '    «+&• 

"■     12a;"     12a;"     12a;" 

4(a;-l)        3(a;-l)  ix 

^''      40«"-l)'     4(a;^-l)'     4(a;''-l)* 

a(a;  +  «)      — a;(a;+«)  a;'  —  aa; 


.294  ELEMENTARY  ALGEBRA. 

fa(a+d)  l{a-h)      ab        V      \  _a{^h)(^y^ 


^^'   {x-iy{x+if  {x-^iy{x+iy'   {x-iy{x+iy' 

4:{x-iy  6{x-l){x+l) 

{x-iy{x-{-iy'  {x-iy{x+iy' 

^^     a{x^  +  ax+a^)      a^—x^         ax 

an      x^+ax-{-a^        x^—ax+a^  a^ 


1 


x'  +  a'x' + a''    x' + a'x' + a''    x' + a'x' + a*' 


Examples — 21. 
a'4-2>'       da''—ab+2¥     26a -20b 


2 


•     2{a-{-b)b'        6{a-b)b    '  12 

ab       a'+b'     a^±b^     g'-gh  +  b^ 
g-y    g-^r    a'-b''        a'-b'    • 


ANSWERS  TO   EXAMPLES.  295 


4.    -^,     0.  '  ^ 


6        " 

"•  4a'- 

■r 

2a; 

a-hhx  ^     l  +  x^  +  x"" 

b  +  ax  '    x^{x'^  +  iy'  '  x+y 

x^  —  y^  '  *  a'{a  +  x)' 

X  —y  a\x—a)  x-\-y 

x-^x^-\-Zx^  l-\-2x-\'':^x^ 


Examples — 22. 

1        '^ 

5^    x{a-{-b)  —  ab 
{x—a){x—by 

(«— a)(«— 5)* 

3             ^ 

4       ^-^-^ 

d*    v/'* 

6                ^ 

7.  1. 

c(c—a)  {c—b) 

Examples — 23. 

1.   1^.              2.  1. 

4. 

1 

296  ELEMENTARY   ALGEBRA. 


5. 

x—a. 

®-      ab    • 

8. 

ax 

9  i^+yy 

x^^-f 

a'-xr 

11. 

X 

abc 

x-y 

14. 

x"    a" 

y' 

A.           15.  1. 

y 

7. 


x^c 


X-\-l) 

^«    x''  —  ((jr''^oJ'x—a^ 

Id.    ~"T"3~^ 


1. 

4. 

d{a-iy 

b{a+b) 

7. 

a  +  x 

cc+y 

10. 

1 

13. 

5.^-1 

ifi 

aj'-Ga" 

2. 


Examples — 24. 
9cV  ^      1 


16a  V  x+y 

^   x{a-^2x)  2x 

a  x—y 

o.  •  "• 


a;— a'  *  c^-a—V 


^-1\'  19   /-^' 


:ca 


2/ 
6^^+a^  +  l  ..    fa^  +  a^)(^Ha') 


17«  .  lo. • 


Examples— 25. 


1.    -.  2.  1.  3.  -^.  4.  x^l. 

X  x^\ 


ANSWERS   TO   EXAMPLES.  297 


5.    1. 

6.  ^-f. 

x—5 

7.  A. 

8.  0. 

9.   i. 

10.    2f. 

11.  0. 

12.  0. 

13.  a. 

Examples — 26. 


1.  w,-27«w^||^,  -^f. 

2.  a;' +  6a;' +  12^ +  8.  3.  :?;'-8^'  +  24:c'*-32a'  +  16. 

4.  iz;' +  lore' +  90:?;'  + 270:?;'  + 405a; +  243. 

5.  l  +  10a;  +  40a;'  +  80a;'  +  80a;'  +  32a;'. 

6.  Sm'-12m'  +  6m-l. 

7.  81a;*  +  108a;' +  54a;' +  1 2a; +1. 

8.  16a;*-32^a;'  +  24aV-8a'a;+a\ 

9.  243a;'  +  810^a;*  +  1080a'a;'  +  720a'a;'  +  24.0a'x  +  32a\ 
10.  64^' -1446^'^*  + 108^5' ~27Z>'. 

n.  aV-3a'xy  +  daxtf-y\ 

12.  aV  +  4f^V  +  6aV  +  4aa;'+a;^ 

13.  32a'm'-80a'm'+80a'm'-40a'm'+10am'-m'^ 

U    a'-da'b+3a'c-h3ad'-6a'bc  +  dac'-b'+db'C'-ddc'+c\ 
15.  l-3a;  +  6a;'-7.^''  +  6a;*-3a;'^  +  a;^ 


298  elementabt  algebea. 

Examples — 27 

1.  4:{aO  +  ad-\-U-^cd).       2.  2(a'  +  2«c  +  c'+Z>'  +  2M+6Z'). 
3.  l+2ir+3^'+2^'  +  a;\  4.  l-2aj  +  3a;'-2a;N-a;*. 

5.  l  +  'Zx-x'-^x'  +  x'.  6.   l  +  6^+13.T'  +  12r^;'  +  4:?;*. 

7.  l-6a;  +  152;'-18^'  +  9a:\  8.   2(4+25^^+16^*). 

9.  l-^x  +  Zx^-x'  +  ^x^-^-x'. 
10.  l-\-^-{-10x'-\-2Qx'-\-^bx'-^'il4.x'  +  l%x\ 

Examples — 28. 


1.  ±:2a¥c\  dtiWy\  ±10a*Z>V. 

.3^^  ^-j^  +^^y 

^-  5^   ^         8a  ^        4a&^' 

aV^    _^'      4^'        Ga^g*^ 
"^^  2    ^        3a;^^       5a*  ^  7   * 


Examples — 29. 

1.  x'+x+l.              2.  l-a;  +  2a;^  3.  a;'+3.T  +  8. 

4.  a;'*-2a;-2.            5.  l-2a;+3aJ^  6.  2a;*-i?;'-2. 

7.  x'-ax  +  2a\        8.  ic''-a2;+S^  9.  a;' - 6a;' +  12a;- 8. 
13 


ANSWEKS  TO  EXAMPLES.  299 


10.  x'-\'2ax^-Wx-a\  11.  l-x  +  x^-x^-^-x'. 

12   |5_^^|^.  13.  1-x,  a-2.  14.  2a-35. 

15.  x''—xi/  +  y\ 


Examples — 30. 

1.  421,  347,  69.4,  737,  1046,  4321. 

2.  2082,  20.92,  1011,  20.22,  129.63. 

3.  1.5811,  44.721,  .54772,  .17320,  10.535,  .03331,  .06324, 

.07071. 

Examples — 31. 

1.  x  +  2y.            2.  a-S,            3.  a;  +  4.  4.  2^-35. 

5.  a+Sb,        6.  2x—7y.         7.  m—4:nx,  8.  ao?— 5Z'2;, 

9.  a'  +  2a  +  l,            10.  a;'-4:z;  +  2.  11.  a'-ab  +  b'. 
12.  a-5+c;. 

Examples — 32. 

1.  21,  23,  25,  32,  4.7,  48,  64,  9.6. 

2.  114,  11.7,  125,  108,  1.41,  192. 
8.  1.357,  .5848,  .2154,  L587. 


300  ELilMENTABY   ALGEBRA. 


Examples— 

33. 

1.  5. 

2.  2. 

3.  3. 

4.  4. 

5. 

-h 

..    d—a 
to. . 

7.  3. 

8.  1. 

9.  4. 

10. 

—\a. 

1.  -4. 

12.  |. 

13.  -|. 

14.  ^'. 

15. 

x=5. 

Examples — 34 

1.  42.         2.  12.         3.  12.  4.  5.          5.  7.          6.  4. 

7.  5.           8.  f.          9.  7.  10.  ^(25a-18^). 

11.  7.  12.  If  13.  11.  14.  5.        15.  2^.      16.  3. 

17.  2.  18.  4.  19.  2. 


Examples — 35. 

1.  10.  2.  8.            3.  12.            4.  6.               5.  -7. 

6.  16.  7.  5.            8.  31            9.  -6.           10.  5. 

11.  8.  12.  J.          13.  3.  14.  2.               15.  7. 

16.  If  17.  i.          18.  1.  19.  17.            20.  2. 

21.  4.  22.  2.          23.  18.  24.  8.  25.  x=2. 

26.  a;=-||.  27.  a;:^-7.           28.  :r=4.  29.  a:=-L 

30.  20.  31.  3.               32.  5.  33.  a- 5. 


ANSWERS   TO   EXAMPLES.  301 


34. 

b—a,            35. 

a-\-b 

37. 

ah 

38.   ^«*,. 

a  +  b 

a+h—c 

40. 

a^l-Vc^d 

41.  c. 

43. 

i{a  +  b  +  3). 

Examples — 36. 

39. 
42. 


a-hb      ' 
a-hb 


b—a 


1.  12.                  2.   9.                  3.   120.  4.  $1.75. 

5.  35,  13.           6.  513,  466.            7.  15.  8.  31,  18. 

9.  15.  10.   90,  60.  11.  IsToYember  20tli. 

12.  16.                13.  37,  30,  20.             14.  20.  15.  41. 

16.  88.                17.  $36,  $12,  $16.  18.  5. 

19.  £45,  £57,  £63,  £65.                       20.  15,  5. 

21.  98f  miles  from  B ;   lOf  hours. 

22,  10,  14,  18,  22,  26,  30.         23.  28,  14.  24.  88,  44. 
25.  5,  6.            26.  22,  7,  12  gallons.  27.  3000. 
28.  18,  3,  3.                     29.  24000.  30.  £140. 

Examples — 37. 

1.  45  gallons.                2.  2450,  196,  98.  3.  84. 

4.  15  feet  by  11  feet.             5.  20  lbs.,  15  lbs.,  15  lbs. 


302  ELEMENTAKY  ALGEBEA. 

6.  $240.  7.  3i  days.  8.  75.  9.  1504. 

10.  1540,  880,  616.  11.  10  lbs. 

12.  18,  lOf,  6i  days.        13.  $1.05,  $1.17.         14.  6f  oz. 

15.  654.    16.  76,30.     17.  21-3^  hrs.,  lOJ^lirs.    18.  12,16. 

19.  10,  15,  3,  60.        20.  240,  180,  144  days.         21.  12. 

22.  20,  80.        23.  5^^.        24.  240.        25.  24.       26.  60. 

27.  25.  28.  7  hours,  5-^',  6  hours,  16^'. 

29.  40  minutes  past  eleven.  30.  $100000000, 

31.  7,  15,  48.  32.  189. 

Examples — 38. 

^      mna  «   m{nb—a)     n{a^ml) 

m+n  *      n—m   ^        n—m 

ma  na 


4. 


m-\-n      m  +  n 

mpa  npa  nqa 


mp+np  +  nq^    mp-i-np  +  nq^     mp-hn^J-^nq 


^      ml—na               ^          dbc  „      d 

5. .  6.  -1 TT-.  7. 


n—m  '   ab  +  ac  +  bc  '    b  +  c 

be 
b  +  c' 


ANSWEES  TO  EXAMPLES.  303 


EXAMPLES-~39. 


1.  10;  7.            2.  17;  19.        3.  2;  13.  4.  4;  1. 

5.  5;  5.              6.  21;  12.         7.  19;  2.  8.  38i;  70. 

9.  6;  12.         10.  fif;  IM-         "•  5;  7.  12.  2^;  1. 

13.  x=l,  y=ri.                               II.  ic=10,  y=24 

15.  a;=144,  2/=:216.              16.  .2;  .2.  17.  10;  8. 

18.  12;  3.            19.  3;  2.          20.  a;  J.  21.  a\  I. 

22.    -^;    -^,.       23.  ^;  ..  24.  ^gr;    -gj. 

25.    -^;   -^.        26.    -^;  0.  27.  «;  I, 

a  +  b^    a  +  b  a+b 


Examples — 40. 

1.  x=l,  y=2,  z=d.  2.  x=7,  y=10,  z=9. 

3.  x=6,  y=6)  z=:7,  4.  x=4:,  y=z—6,  z=6. 

5.  a;=3-5,  y=6,  z=-2,  6.  a;=l|,  i/=2f,  ^=:-12. 

7.  x=:2,  y=-dy  z=:L  8.  a:=12,  y=12,  z=l2. 

9.x=6,y=:7,z=-3.  10.  |;  f;  f. 

11.  a;=i(J  +  c--a),  &c.  12.  x=%{a  +  b  +  c)-'a,  &c. 

13.    a:=K^  +  c),&c.  14.    x=y=z=:-      ^^^ 


^ab  +  bc+ca 


304  ELEMENTARY   ALGEBRA. 


Examples — 41. 


1.  ^5g.       2.  48.       3.  108  sq.  ft.       4.  4  liours,  6  hours. 

5.  20,  30,  60.  6.  24,  72.  7.  49;  21. 

8.  45;  63.  9.  |.  10.   (24-1)20. 

11.  1 ;  2.  12.  50  yards;  rates  4  and  5  yards  per  minute. 
13.  11,  and  5,  gallons.  14.  A.  D.  1752.  15.  50;  75. 
16.  90;  72;  60.  17.  4;  2.  18.  8;  5. 

19.  4  miles  walking,  3  miles  rowing,  at  first. 

20.  30 ;  50  miles  per  hour. 

21.  60  miles;  passenger  train  30  miles  per  hour. 

22.  150;  120;  90.  23.  rr=40,  ^=160,  ^^=480. 
a-\-b     a—b 


24. 
25. 


771C — all  +  am. {n—h)  ^      i7i—nc  +  dn{a—7n) 
rob— an  ^  mh—an 

l  +  a*      a^—h  2n  2n 


2a  '       2a   '  '   m—V     m  +  1* 

Examples — 42. 

1.  x^-\-x^,  +2:^,  +a:3;  ah"^  +  ah^  +  ah"" -^  a^. 

2.  ah^-\-a^-Va'b^+ah^)  ah'''\-al)'+ah'  +  ah\ 


ANSWERS   TO   EXAMPLES.  305 

a'b-'  +  3a'd-'  +  6ab-'+4.a-'b  +  2a-^b^ ; 

1    1    A  _i_  _A_ 

a'^  b''^  c''^  a-'b^  ab-^' 

1  3  5  4  2 

a-'b''^  a-'b^ a-'b'^ a'b-^^ a'b-'' 

1  4  2  1 

^^      3a-^Z>^c^  ^  a'b  c-'  '^  ab-'c~'  ^  3abc ' 

1  2  3  5 

'^      3.-,       o         o     '       .      S.7I.2+        ,4 


5.  v/^+2V«'^+3Va'+4V6x  +  Va', 

Va    V{a:'b)      2V(ac')     V{Fc')     Vjbc') 
Vb''^    2v/c  "^    3Vb'    "^    4Va  "^5V^'' 

^'     a'^  b^'^abc'^  aW  '        Va''^  Vb'  "^  ^fa'  ^Vb'' 

a  6>  be  a       vb      vb        Va 


Examples — 43. 

1.  i.  2.1  3.^.  4.100.  5.^. 

6.  ^-^  7.  a^  8.  ^-^  9.  a-\  10.  a^"^- 

11.  .7:^-^1.  12.  a-b.  13.  .T^+2.'z;^  +  a:-4 


S06  ELEMENTARY  ALGEBRA. 

14.  x'+l+x-\       15.  a-'-l.         18.  a'-3a^  +  da-^-a--\ 
17.  a''  +  2ahi  +  ab-x^yK  18.  x^+x^y^  +  x^y^  +  y^. 

19.  a^  +  t«^^^+^^  20.  lex-^-nx-^y-^^+dy-^' 

21.  a;+2/.  22.  a^-ah^+hK  23.  a*+2>^-c*- 

24.  a;*  +  2iz;M+3a;*a+2A^-}-^'. 

25.  x^-2x"K  26.  ^r-2-ar\         27.  aJ"*  + 1  +  a-^Z». 

Examples — i4. 

1.  64^  81*    {i)K  {i)\  {i)K  8i 

2.  25^,  (V)*,  (K)^    (K)i,    {i(^^+2a^+J^)}i; 

125^  (1^)4,  (^a^)^,  ftW)^  {i(a^+3a^^+3«&»+J')}i 

6561-^  (n^3^)-%    (a«)-^  (^)"*. 

4.  v/125,  v/3,  v/12,  v/|,  x/i   v/320. 

5.  V54,  V256,  V2048,  V3,  V|,  V^. 

6.  v/(4a),  v/(98a^a;),  j/^. 

7.  v/(2a5),   v/(6a':.),   |/||;,     ^^,     .^{^'-^r'). 


ANSWERS   TO   EXAMPLES.  307 

8.  3v/5,  5v/5,  36v/3,  3V5,  18V2,  iv/G,  V12,  V54,  6. 

9.  4V2,  8V2,  6V48,  fv/2,  ^v/2,   fV2,   ■|v/21,   |V150, 
V375. 

10.  2v/3,  15v/3,  |v/3,  ^n/3,  iv/3,  ^VB. 

Examples — 45. 

1.  v/108,  v/112;  V81,  V80;  V120,  V128,  V135. 

2.  7v/2.  3.  9V4.  4.  |v/3.  5.  '^. 

9.  2+|n/6. 

10.  iCv/S+v/S+v/S),  iv/6+iV33+iV120. 

11.  ^(2y2+v/3),   v/5  +  1,  v/5-v/2,  4+v/2,  i(7+3v/5). 


Examples — 46. 

1.    v/3+1.  2.  3+v/2.  3.    v/5-v/3. 

4.  2V6-3V2,  5.  4v/2-3.  6.  iV6-l. 

7.  2-^|v/3. 


308  ELEMENTARY  ALGEBRA, 

Examples — i7. 
r  4  2.  50.  3.  25.  4.  H.  5.    {a--h)\ 

(,.  a.  7.  -^^.  8.  -3^3^. 


Examples — 48. 

1.  ±3. 

2.  ±3.                3. 

±1. 

4.  ±4. 

5.  ±i. 

6.  ±2^.               7. 

±|. 

8.  ±5. 

9.  ±3. 

10.  ±5.               U. 

±2. 

12.  ±2. 

13.  ±%/3. 

14.   £B=d=3. 

Examples — 49. 

1.  4,  -2. 

2.  -1,  -9. 

3. 

20,  -6. 

4.  7,  5. 

5.  8,  -40. 
Examples— 50. 

6. 

10,  -110. 

1.  1,  -8.  2.  17,  -4.  3.  —5,  -20. 

4.  -1,  -12.  5.  1,  -20.  6.  25,  -136. 


answers  to  examples.  309 

Examples — 51. 

I.  0,  -51.  2.  6,  -ii.  3.  8|,  -10. 

4.  14,  -lOf.  5.  12,  -12tJj.  6.  13,  -11^ 

Examples— 52, 

1.  10,  2.  2.  3,  -1.  3.  2,  -I 

4.  U,  -i|.  5.  If,  -n.  6.  7,  -li. 

7.  3,  i-  8.  i(-9±3v/3).  9.  3,  if. 

10.  3,  -f  11.  |(37±V57).  12.  2,  -3. 

Examples — 53. 

1.  G,  3^V        2-  6,  -4|.        3.  1,  lOf.  4.  3,  -8,'^. 

5.  5,  -l^-V  6.  5,  li.  7.  5,  -li. 

8.  31,  0.        •    9.  a±-.  10.  (a±by. 

a 


310  ELEMENTARY   ALGEBRA. 


Examples — 54. 

1.  x'-'ke-21= 

:0.                        2.  6a;''+5a;- 

-6= 

0. 

3.  cc^'+llx+SO: 

=0.                       4.  Sx^-Sx-- 

=0. 

5.  a;'-100=0. 

6.  a;'-3aa;+fl'- 

-4=0. 

a 

7.  a;''  +  2a;-l=0. 

EXAMPLES — 55. 

• 

1.  ±2,  ±3. 

2.  49.              3.  4 

4.  ±4, 

5.  5,  -3. 

C.  3,  -2. 

7. 

12,  -3 

8.  9,  -12, 

9.  ±3.            10.  2. 

11. 

4. 

12.  16. 

13.  1,  |.           14.  4. 

15. 

3a'. 

16.  0,  ±5. 

17.  0,  ±V2. 

18. 

2,  ±1. 

19.  0,  ±v/(a5). 

20.  a,  -2flr,  —2a. 
EXAMTLES— 56. 

1.  3,  4,  5.        2.  36,  24.        3.  30,  24.  4.  18,  12,  9. 

5.  196.  6.  ±12,  ±15.  7.  24. 

8,  15  yards,  25  yards.        9.  4550.        10.  40  yds.  by  24 


ANSWERS   TO   EXAMPLES.  311 

11.  16.  12.  4  yards,  5  yards.  13.  £60,  or  £40. 

14.  8d.  15.  Equal. 

Examples — 57. 

1.  a;=7,  ?/=±4.  2.     x=4:,  y=-dj 

a;=-3,  y=4.  ^ 

S.  a;=4,   ^=3,)  4.      cc=:8,  ^/^^^^ 

5.  x=:6,  y=5, )  6.         a:=5,  y=^,) 

7.  z=:5,  y=3j  8.        cc=3,  y=4:J 

9.  a;=4,  2/=^J  10-     ^=10,  y=15,^ 

x=2,  y=4.5  a;=-.10|,  y=-16i.\ 

11.  a;=3,  ^=2,^  12.         a:=5,  2/=4,) 

t;=4,  y=5.$ 


13.  a;=i{aiFv/(25'-a'')},i 


14.  x=i{±V{W  +  b')  +  I>},-i 


U.  x=8,y=l,-i       ,„         ^        «' 

=1,  y=.8.i     ^^'  ''=^T777;rn?i'  y=^- 


x=l,  i/=8.  $  v'(a''  +  5^)'  ^~    ^{a'  +  d'Y 


312  ELEMENTARY  ALGEBRA. 


18.  fl^  +  Z>  +  l, --7—;    I?,- 


a-hl    '      '    a  +  1 


19.  ±-|;     ±3Z>.  20.  ±1-;     ±2^. 

d  4: 

21.  0,  a-Vh,  h{a-l))^hV {{a-Vdh){a-l))}. 


Examples — 58. 

1.  11;  7.        2.  8;  24.        3.  10;  12.  4.  18-  8:  6;  16, 
5.  5;  3.         6.  4;  2.           7.  2;  2.  8.  ?;  4. 

9.  60.  10.  6,  .4.  11.  160;  £2. 

12.  756;  36;  27.        13.  £275,  £225.  14.  2,  5,  8. 


Examples — 59. 


*•  ia?  io  ^  irf^ 

TFt  5 

mz>  ihh,  AW 

2.  «+J. 

*'-"363- 

^-                        4             .4 

-    »'— lla;  +  28 

4.  A,  |. 


7.  1. 


ANSWERS   TO    EXAMPLES.  313 


8.  -('iti)"  9.  i^l^.  IJ.  c.  8. 

o{a  —  o)  o—d 

12.    35,  42.  13.  4.  14.   — ^. 

a-\-o 


Examples — 61. 


1.  10,  4|,  2tV  2.   9,  ^,  If.  3.  6,  1|,  If. 


10. 


^(^1)  '         ^^    a  +  h,OY  h{a-h).        12.  .^'  =  l,  y-=i\ 


13.  rr:=±9,  //=±3.  14.  3.  15.  25,  20. 

16.  8:7.  17.  6. 

Examples — 62. 

1.  32,  272.  2.  39,  400.  3.  63,  363. 

4.  694,  34750.  5.  9,  IG.  6.-1,  0.          7.    -28. 

8.  -275.  9.  16i.  10.  -m.         11.  336f. 

12.  -84. 

Examples— 63. 

1    12.  2.  20.  3.  I^Vt^  1-^V^  &c.,  &c.  ;  6  =  60. 

4.  14,  16,  18.  5.  141    14|,...  6.  6J,  5,... 


314  ELEMENTARY   ALGEBRA. 

7.    -h  h'-     8.  10,  4.          9.  82.  10.  5,  9,  13,  17. 

11.  1,  2,  3,  4,  5,           12.  18,  19.  13.   7.          14.  5. 

Examples — 64. 

1.  64,  85.                   2.   1280,  1705.  3.   96,  189. 

4.    __256,  -170.         5.  4096,  3277.  6.    -512,  -341. 

T.  tV^.            8.  1H|.            9.  4,VV  10.  2f-||. 


Examples — 65. 

1.  8.  2.  H.  3.  i  4.   ^.  5.  I 

6.  4.  7.  tV  8.  1.  9.  1/,.        10.  h. 

11.  iV^.       12.  lU. 

Examples — 66. 


I.  4,  16,  64.       2.  8,  12,  18,  27.       3.  -9,  27,  -81,  243. 

4.  3,  12,  48;  or  81,  -54,  36.  5.  1,  3,  9,  .  . 

14 


ANSWERS   TO   EXAMPLES.  S15 

6.  ^^.  7.  2  +  |  +  f  +  &c.  ;  or  4-|  +  |-&c 

8.  3_|4.|_&c.         9.  200  miles. 

Examples — 67. 

1.  iil.  2.  f,  A,  2.  3.  3,i^. 

4.  A.  tV.  a.  5.  6,  12.  6.  36,  64 

7.  1,  0.  8.  3,  9. 

Examples— 68. 
1.  720,  720.  2.  5040.  3.  19958400. 

4.  34650.  5.  210.  6.  6.  7.  4.  8.  6, 

Examples— 69. 

I.  126,  84,  36.  2.  330,  330,  11.  3.  3003,  455. 

I.  4950.  5.  210,  84.  6.  IL 

7.  50063860,  5006386.  a  116280. 

Examples — 70. 

1.  l  +  Gx+lox'  +  20x' -^iDx' -\-6x' -hx', 

2.  a'-loa*x  +  dOaV'-270a'x''^4:06ax*-24Saf. 


316  ELEMENTARY   ALGEBRA. 

4.  a'-9a'a;  +  36aV~84aV  +  i2GaV-126aV-f  84aV 

-36aV  +  9ax''-x^ 

5.  1  +  12x-{-66x'-{-220x'  +  ^9dx'  +  792x  +  924a;"  +  792a;' 

'\'4.mx'+220x'  +  6Gx''  -\-12x''  +x'\ 

6.  l-20a;-fl80a;^-9G0a;^4-33G0x*-8064a;^  +  13440a;^ 

-15360a;'  +  11520a;''-5120a;'  +  1024a;^°. 

7.  ^•-18r/'a;  +  135c'^V-540aV  +  1215aV-14o8aa;* 

+  729a;". 

8.  25Ga;^  +  1024r/a;'  +  1792aV  4-  1792aV  +  1120aV 

+  448a'^a;'  +  112aV  +  lea'x-^-a', 
a  128a'-1344a«a;  +  6048r^V-15120aV  +  22680«.V 

--20412r^V  +  10206aa;''-2187a;^ 
10.  l-^6x■i■\^-x''--16x'  +  ^^x'-\^x''  +  ^%^-x''-^x''  +  ^,d' 

U.  l-i^x  +  %^x^^^i-x'  +  ^\^x'-ii^x'^iUx'-¥A^' 

12.  36aV.  13.  -l^^XAVy. 

110 

14.  4t05a"b'.  13.  t^=^V, 

[5  |o_ 


answers  to  examples.  317 

Examples — 71. 

1.  100101100,  102010,  10230,  2200,  1220. 

2.  41104,  23420,  14641,  7571,  5954. 

3.  402854.     1  511,  22154.     6.  a 

Examples— 72. 

1.  152.  4.  100001000000  (binary)  ==  201000  (quat.). 

5.  57264,  95494,  eltS.  6.  4112,  6543,  62/^. 

7.  1295,  216;  2400,  343;  4095,  512. 

Examples — 73. 

1.  .9030900,  .9542426,  1.0791813,    1.3010300,  1.3979400, 

1.7781513. 

2.  r.5228787,  1.3979400, 1.6020600,  2.4771213, 
2^5228787,     375185140. 

3.  2.2253093,  .0170334,  3.5670265. 


318 


ELEMENTARY   ALGEBRA. 


4.  2  + log.  3  +  2  log.  7. 

5.  Gh-2  log.  3  +  3  log.  .21. 

6.  2  log.  2-t  log.  3  +  f  log.  5-1,  and  I  log.  13 -|  log.  2 

7.  .019.  8.  4  and  6.  9. 1.8035700. 


'  1 


_</x 


^. 


) 


A* 


^- 


P"^^  V  J 


V  .^^ 


x./    f  rn  ^    If 

U.  C.  BERKELEY  LIBRARIES  ^        %^ 

iiiiiiiiii  nil  nil  nil  III  II  nil  mil  mil  III  III!       ^.v^  ^. 


^\. 


s    %, 


